Tag Archives: card counting

Harvey Dubner & the Development of the Hi-Lo Strategy

Basic HI-LO StrategyBy Leslie M. Golden

Reprinted with permission.
Originally published as Appendix D in the
book Never Split Tens by Leslie M. Golden

Harvey Alan Dubner was born on July 14, 1928, the son of Reuben and Frances Dubner, Jews of Eastern European descent who lived in New York.  He had a younger brother, Neil Peter Dubner.  He and his wife Harriet (Weiss) had four children, in order of age, Robert Joseph, Emily Rachel, Terry Ann, and Douglas David.  Although his entire family was Jewish, his son Robert’s recollections below indicate they did celebrate the exchange of gift ritual of Christmas with Hanukkah.

He attended Brooklyn Polytechnic University in New York, from which he earned the Bachelor’s of Science in electrical engineering in 1949, summa cum laude, and the Master’s of electrical engineering in 1951.  He played flippantly with his middle name.  Although his birth certificate states his middle name as “Alan,” his Master’s degree diploma from Brooklyn Polytech in 1951 provides it as “Allen.”  His son Robert notes that “it is clear that he never knew how his middle name was spelled.  He would use his middle initial when forced to. But asked about his middle name, he would shrug and spell out Allen.  But it was always clear to me that he was just coming up with something to make the questioner go away; he neither knew nor cared how it was spelled.” 

Although educated and employed as an electrical engineer, he was a proficient mathematician and as early as the 1950’s a skilled computer programmer.  He found great pleasure and joy in attacking mathematical problems working alone.  He would first try to solve the problem analytically, using, that is, only equations.  If the problem was not amenable to such solution, he would write a computer program to simulate the effect.

His career brought him to in 1960 to the Curtiss-Wright Corporation in Paterson, New Jersey.  There he used an IBM 1620 computer to run blackjack simulations, leading to the Hi-Lo strategy.  In so doing, he solved the problem of an optimum bet, independently recreating the 1956 work of John L. Kelly, now known as the Kelly criterion.

He considered the 2’s through 6’s as “lows” and the ten-value cards and Aces as “highs.”   His analysis then showed that an index calculated by subtracting the number of low-value cards left in the deck from the number of high-value cards left in the deck and dividing by the number of cards remaining to be played provided an excellent sensitivity to the quality of the cards remaining to be played.

Uproar at the 1963 Fall Joint Computer Conference

Dubner, then 35 years old, presented his blackjack results during a panel discussion on games of chance and skill, entitled “Using Computers in Games of Chance and Skill,” moderated by Edward O. Thorp, at the 1963 Fall Joint Computer Conference held in November at the Las Vegas Convention Center.  He had learned of the panel discussion, most likely in the trade magazine, Electrical Engineering, and had contacted the conference and asked to be included as one of the presenters.

A “call for papers” had been published on page 51A of the April, 1963, issue of Electrical Engineering magazine, the trade magazine of the IEEE, the Institute of Electrical and Electronics Engineers.  (I am indebted to Ms. Jennifer Hart of the Crerar Library of the University of Chicago for her aid in locating this announcement.)  It was general in nature, and did not mention the panel discussion on games of skill and chance.  The actual program, a listing of session topics, speakers, and their topics, was published in the September, 1963, issue.  It mentioned the panel discussion, but did not mention Thorp’s name in that context.  The call for papers and the program, including the panel discussion, may well have been published elsewhere. 

Gambling writer and player Jerry Patterson, who writes that he was a speaker at the panel discussion, relates in his column “Harvey Dubner:  The Forgotten Man of Blackjack” that the conference organizers not realizing that most “computerniks” (as they were called at that time borrowing the suffix from the Russian Sputnik satellite) were also inveterate blackjack players, they scheduled the Panel Discussion in one of their smaller meeting rooms. The room filled up and overflowed 45 minutes before the session was scheduled to start!  Hundreds of conference attendees were pushing and shoving to get into the room.   The crowd, of course, had been drawn by Thorp. 

Newspaper coverage of the conference provides insight into Dubner’s sense of play.  He asked the panel, “Can you make money playing blackjack?”  After the panel emphatically replied, “Yes,” Dubner said, “Then I wish we’d get this meeting over with.  I’m losing $25 an hour while I’m sitting here.”

Dubner had his results for his betting and playing strategies printed on 3 x 5 index cards.  The clamor by those in the overflow, standing-room only crowd to obtain a copy of the “Basic HI-LO Strategy” card caused, in the words of his wife, Harriet, a “riot.”  Jerry Patterson writes that Dubner “stole the show” and was “mobbed” at the conclusion of the panel discussion by the crowd “all wanting copies of his handout.”  Patterson recalls, “Here at last, many were saying, is a system that is practical, that can actually be used in the real world of casino play.” 

An image of that card is presented in Figure 1.  Figures 2 and 3 present additional material found by the Dubner family folded neatly in Dubner’s copies of the 1962 and 1966 editions of Beat the Dealer.  A total of nine graphs, tables, and summary of analyses were found copied onto the back and front of 8 1/2 x 11 sheets of paper.  These were most likely copies of slides he presented at the panel discussion or handouts for attendees at the panel discussion.  Of course, the material could have both been presented as slides with copies reproduced for handouts.  Thorp remembers they were presented at the panel.  Dubner’s son Robert doubts that they were presented at a later meeting or published after the conference.  The similarity in font and styles of the various materials implies they were created at about the same time, for presentation together.

His graph #5 is entitled “Frequency of Favorable Situations for High-Low System, Reshuffle at 6 Cards from End.”  This implies that he not only used the terms “Hi-Lo strategy” and “High-Low system” somewhat interchangeably, with comments by his son Robert reproduced below indicating a preference for the former, but also that Thorp had probably borrowed the term “high-low” to use in his 1966 revised edition of Beat the Dealer.

Basic HI-LO Strategy

Basic HI-LO Strategy

Figure 1.  Dubner’s Hi-Lo strategy as provided to the panel discussion on “Computers Applied to Games of Skill and Chance” at the 1963 Fall Joint Computer Conference.  The top image is that of an actual 3 x 5 card that was handed out to attendees.  The clamor by attendees to obtain one of the copies caused, in the words of Dubner’s wife, Harriet, a “riot.”  The bottom image is taken from an 8 1/2 x 11 photocopy of nine apparent slides most likely presented at the panel discussion.  Little doubt remains that these materials, either as slides and/or as handouts, were in fact presented at the panel discussion.  Thorp remembers that they were.  Dubner’s son Robert doubts that they were presented at a later meeting or published after the conference.  (Courtesy of Dubner family)

Following the presentation, Thorp and Julian Braun embarked on a detailed analysis of the Hi-Lo strategy, Braun using the computer of the IBM facility in Chicago and the programming approach Thorp had used to develop the ten-count strategy he had introduced in the 1962 edition of Beat the Dealer.  Thorp published the system in the 1966 revised edition of Beat the Dealer as the “complete point count system.”  Some refer to it as the “High-Low” system, a term Dubner introduced, as noted above.

Although Jerry Patterson pointedly asks the rhetorical question, “Did he (Thorp) have Dubner’s permission?” such a request is not normally needed in scholarly work, and Thorp fully acknowledged Dubner’s work in the 1966 revised edition of Beat the Dealer.  Perhaps the more interesting question concerns why Thorp and Braun excluded Dubner from this additional work.  We can only speculate on the answer.  Perhaps Dubner, a private man, simply wanted to return to his professional endeavors, considering the blackjack calculations an interesting diversion now solved to his satisfaction.  Perhaps Thorp was annoyed that Dubner “stole the show.”  I hope and believe that the former was the case. 

Return on Inventment Theory

Figure 2.  Dubner’s summary of his consideration of betting in terms of financial theory.  The image is taken from a photocopy of nine apparent slides most likely presented at the panel discussion on “Computers Applied to Games of Skill and Chance” at the 1963 Fall Joint Computer Conference.  Little doubt remains that these materials, either as slides and/or as handouts, were in fact presented at the panel discussion.  (Courtesy of Dubner family)

It was natural for Thorp to compare the power of the complete point count, based on the Hi-Lo strategy of Dubner, to his ten-count strategy.  Thorp, on page 94 of the 1966 revised edition of Beat the Dealer, opines that the two systems are of “comparable power.”  Dubner created a comparison of the Hi-Lo strategy to the ten-count strategy, which may have been presented at the panel discussion.  Thorp recalls that it most likely was presented.  The results are summarized in Figure 3, the only known rigorous comparison of the two strategies.  Dubner’s analysis shows that the Hi-Lo is superior.  The complete point count system, more detailed than the Hi-Lo, would have been even more powerful.

Jerry Patterson suggests that without Dubner’s development blackjack would not have attained its popularity as the most popular casino game.  The ten-count strategy presented by Thorp in the 1962 edition of Beat the Dealer and in the 1966 revised edition requires the player to calculate an index including the tenths and hundredths digits.  This cumbersome necessity in his view limits the number of players able to play blackjack profitably.  In Patterson’s view, the development of the easier-to-use  Hi-Lo  strategy  by  Dubner  is   responsible  for   the  wide popularity of the field of card counting.  Inexplicably, although my conversations with the Dubner family make it clear that his joy resulted from solving the problem rather than recognition by peers or others, Dubner has not been elected to the Blackjack Hall of Fame.  To me this is an egregious oversight and I hope it is rectified during his lifetime.  Without Dubner, no Blackjack Hall of Fame would exist.

Editor Note: Unfortunately, Harvey Dubner
passed away in 2019, about two years
after this article was written.

Examination of Family Claim

Working with the Dubner family, we can discuss the question of whether Dubner indeed developed the Hi-Lo strategy without, as they claim, ever hearing of Thorp.  The recollection of events more than fifty years in the past by a man near 90 years of age is challenging.

Comparison of Hi-Lo and Ten-Count Strategies

Figure 3.  Dubner’s comparison of his Hi-Lo strategy and Thorp’s ten-count strategy.  The image is taken from a photocopy of nine apparent slides most likely presented at the panel discussion on “Computers Applied to Games of Skill and Chance” at the 1963 Fall Joint Computer Conference.  Little doubt remains that these materials, either as slides and/or as handouts, were in fact presented at the panel discussion.  This provides the results of the only known rigorous comparison of the two strategies.  Thorp, on page 94 of the 1966 revised edition of Beat the Dealer, opines that the two strategies are of “comparable power.”  Dubner’s analysis shows that the Hi-Lo is superior.  The “Figure of Merit” is defined in Figure 2.  (Courtesy of Dubner family)

Most, if not all, blackjack historians have assumed that Dubner read the 1962 edition of Thorp’s Beat the Dealer and was thereby stimulated to examine the game himself, leading to the his development of the Hi-Lo strategy.  Dubner’s son, Robert Dubner, in our early correspondences made the same assumption.  It was natural to do so. 

Thorp distinguished tens and non-tens in his ten-count strategy published in 1962 and Dubner distinguishing “highs” vs. “lows” in the “Hi-Lo” strategy seems a natural extension rather than a complete re-invention. Yet, further discussion with those members of the Dubner family involved, Harvey, his wife Harriet, and his eldest son Robert, has led them to change their opinion.  Dubner firmly states he did not know of Thorp until the 1963 conference.  That he worked independently of anyone else on the project is well-known, confirmed by Thorp himself.

An absolute validation of this thesis would be had if the Dubner family retained some of the paper tape  printout of  Dubner’s  computer runs.   In the early days of computers, output was provided on paper tape and punched cards.   Later computers provided  output on  magnetic tape and within the computer memory.  Those paper tape printouts would likely have either had the date of the run encoded or Dubner would have identified them by marking them with a marking pen, stick-on label, or a note taped to the paper tape.  A date of any run pre-dating the November, 1962, publication of Beat the Dealer would establish with certainty that Dubner developed or had begun to develop the Hi-Lo strategy without any external influence.  As we see below, however, the tapes had been discarded by the family.

Much evidence suggests that Dubner did know of Thorp’s work.  As noted, the documents discovered by the Dubner family folded in his copies of Beat the Dealer seem to be copies of nine slides created for presentation at the conference or publication and/or handouts of such for the conference.  They include Figure 3, a comparison of his work with Thorp’s.  Dubner possesses copies of both the 1962 and 1966 editions of Beat the Dealer.  He used the term “strategy” to refer to his system, the same term that Thorp uses in the 1962 edition to describe his five-count strategy and ten-count strategy.  Dubner calculates a “Hi-Lo Index,” shown in Figure 1, and Thorp calculated a “ten richness” index by calculating the ratio of “others/tens,” although he doesn’t use the term “index,” referring to it simply as the “ratio.”  Both men provided not only a guide to playing strategy but also a guide to betting.  Thorp based his on the Kelly criterion (see later).  Dubner apparently did so as well, but determined the criterion by himself.  No previous system, including that of the Four Horsemen of Aberdeen, had provided a betting guide.

On the other hand, the work was apparently performed while Dubner was working at Curtiss-Wright in 1962–1963.  If the work was begun prior to the November, 1962, publication of the 1962 edition of Beat the Dealer, then Thorp was not a seminal influence.  If it was begun in late 1962 or 1963, then whether or not Dubner had become aware of Thorp’s work remains in question.

The evidence, accordingly, seems to be that Dubner’s recollection is incorrect. 

The alternative possibility, however, remains, that he developed the Hi-Lo strategy before learning of Thorp.  As will be discussed later, Dubner had an interest in playing blackjack in Las Vegas long before the 1963 conference.  With the development of computers, Dubner became a skilled programmer and it is conceivable, as the family states, that he created software simulations of the game for diversion.  He had access to computers from his professional work.  As an electrical engineer, he either subscribed to the trade magazine Electrical Engineering or read it at work.  A call for papers for the 1963 Fall Joint Computer Conference was published in that journal in April, 1963, and the program was published in that journal in September, 1963, including a mention of the panel discussion on “Using Computers in Games of Chance and Skill.”  Perhaps the call and program were also published elsewhere. 

We need to remember that no journals dedicated to “computer science” existed at that time and indeed many early computer science curricula were part of university departments in “electrical engineering and computer science.”  Computer science in its early years concerned the electronics and electrical components more than the software.  I remember as an undergraduate at Cornell University in the 1960’s courses being designated by “EECS.”  It could have been that Dubner developed or began to develop the Hi-Lo strategy and then read of a forum in Electrical Engineering at which to provide his results, without ever hearing of Thorp and his work.

Although neither the call for papers nor the program as published in Electrical Engineering mentioned Thorp’s name, that Thorp was to moderate the panel had to have been publicized somewhere.  As Jerry Patterson pointed out, Thorp was “the draw” and the conference organizers must have utilized the renown he had achieved resulting from the 1962 edition of Beat the Dealer to arouse interest in the panel discussion.  The overflow crowd attests to their success. 

At the date of this writing, Thorp has not been able to provide information from his files as to when he was invited to moderate the panel, when and how that development was publicized, and if he selected the members of the panel.  It seems highly likely that he did have a major impact on selection of the panel.  Of the participants noted on pages 93–94 of the 1966 revised edition of Beat the Dealer, he had worked with Julian Braun, William E. Walden was his student at New Mexico State and later a collaborator, and Allan Wilson apparently knew Julian Braun from San Diego State College.

Dubner either then learned of Thorp or did not learn of Thorp.  In the former case, Dubner, now committed to participating in the panel, would have learned of Thorp and would have purchased Beat the Dealer.  He could have then run simulations of the ten-count strategy using the same software that he had used in analyzing the Hi-Lo strategy and then prepared what is apparently a slide, reproduced in Figure 3.  In this scenario, he developed or had begun to develop the Hi-Lo strategy before becoming aware of Thorp.  He may have added the Hi-Lo Index and betting guide afterwards.

On the other hand, it may be that Dubner remained unaware of Thorp’s work until the conference itself.  Crucially, the September, 1963, issue of Electrical Engineering which presented the program for the November conference provided “R. A. Kudlich” as the chairman of the panel discussion, without mentioning Thorp’s name in that context.  The call for papers did not even mention the panel discussion.  Perhaps the conference organizers publicized Thorp as moderating the panel in other journals which Dubner had not read and/or by poster flyers mailed to academic departments.  In this scenario, Dubner learned of Thorp at the conference, bought Beat the Dealer after the conference, performed the analysis that led to the results of Figure 3, and then either presented the nine apparent slides at a later meeting or published them.  He may have simply held onto them himself. 

This scenario is unlikely.  The Dubner family states that Dubner did not present the information in the nine apparent slides at a later date, and Thorp recalls that Dubner presented the information comparing the Hi-Lo to the ten-count strategy at the panel discussion.  In fact, on page 94 of the 1966 revised edition of Beat the Dealer, Thorp writes that Dubner made “great claims” for his strategy and that “(H)is calculations supported his claims.”  This implies that Thorp’s recollection is correct, that Dubner presented the results of his computations.

One piece of intrigue concerns the comment by Thorp on page 94 of the 1966 revised edition of Beat the Dealer.  As noted above, there Thorp states that the Hi-Lo and ten-count strategies are of “comparable power.”  Figure 3 shows that the Hi-Lo, even in Dubner’s formulation, is superior.  The more detailed calculations subsequently made by Julian Braun would have produced an improvement, in particular, providing strategy decisions for all player strategies, not only the hard standing guide provided by Dubner.  Thorp’s statement implies either an emotionally-motivated defense for his ten-count strategy, which is understandable but which I hope is not the case, or that Dubner did not in fact produce the comparison slide at the conference (and that Julian Braun did not later compare the two).  The latter case would imply Dubner bought Beat the Dealer after the conference, supporting Dubner’s recollection.  Thorp, as noted, does recall that the comparison information was in fact presented.

As noted, Thorp based his betting guide on the optimum bet analysis of John L. Kelly of Bell Telephone Laboratories, commonly referred to as Bell Labs, published in 1956.  Robert Dubner believes that his father independently derived the result.  I do believe that belief to be plausible, Dubner’s mode of scholarly work being to work independently of others.  (In the sciences, the first step taken by researchers is to go to the library, now usually digital but in previous generations the physical library, and determine if the work has been done by others.  In mathematics, an incredibly splintered field, perhaps this is not always the case.) 

The Kelly criterion, which can be rigorously proven to provide the optimum betting size, is simple and intuitive and can be expressed for the case in which only two outcomes from a $1 bet are possible as

f = pq/b ,

where f is the optimum fraction of your current bankroll to wager, p is the probability of success, q is the probability of failure, and b is the net bet or payoff, the amount of money you win if successful above and beyond your $1 bet.  (If, for example, you are playing a game with a 0.60 probability of winning, therefore a 0.40 probability of losing, and a payout of $2 for every $1 wagered, then the optimum fraction of the bankroll to wager is 0.40.  Go for it!!)  From the formula, we see that the optimum bet increases with increasing probability of success, decreases with increasing probability of failure, and increases with increasing size of the payout.  All dependencies make sense.  It is often equivalently expressed as

f = (bpq)/b,

with the numerator then being the expectation value of a single $1 bet and the denominator being the payoff for a winning $1 bet.  I’m sorry, although in this form we can refer to the criterion in pithy terms as being “expected winnings/winnings from successful bet,” I prefer my algebraically simpler version.

In addition to the rigorous proof, numerous “heuristic,” that is, reasonable, practical demonstrations, of this simple formula exist.  It is reasonable that Dubner, a gifted mathematician, was able to prove the formula himself without knowledge of Kelly’s work, or to simply derive it by trial and error and computer simulation using the three parameters, p, q, and b, in different combinations.  After all, the above form, bpq, and (pq)/b are some of the only few simple algebraic forms in which p, q, and b behave in the sensible manner noted above.  Indeed, mathematician and physicist Daniel Bernoulli (1700–1782) had effectively arrived at the formula based on an analysis of geometric means of possible outcomes centuries earlier, and his result was commonly known among mathematicians as well as economists.  Deriving it required a statement of the problem, a basic knowledge of probability theory, and use of the calculus, within the skill set of most mathematicians.

After considering these various possibilities, my feeling is that Dubner began and performed the majority of his investigations without hearing of Thorp.  At some time, perhaps after the call for papers or after reading the program in the September, 1963, issue of Electrical Engineering, he learned of Thorp.  He then created the comparison diagram and presented it to the panel discussion along with his other material.  The evidence does not support Dubner learning of Thorp only at the conference itself.

In research, investigators build upon the work of their predecessors.  Thorp was initially influenced by the work of Baldwin, Cantey, Maisel, and McDermott.  Dubner’s work was at some stage influenced by that of Thorp.  Thorp’s work subsequent to the 1963 Fall Joint Computer Conference was influenced by the work of Dubner.  That one researcher builds upon the work of others is how scientific knowledge progresses.   No one’s work is diminished in importance as a result.  As Sir Isaac Newton wrote (in a phrase the first documented form of which was stated by the 12th century scholar Bernard of Chartres), “If I have seen further it is by standing on the shoulders of Giants.”

Dubner was always and remains a quiet man, not interested in fame or fortune.  His interest was mainly in the joy of solving mathematical and engineering problems.  Like many of us interested in the mathematics of the game, he was not at all a gambler.  It is appropriate that the lack of biographical material on his life, personal philosophy, research approach, and contribution to the game of blackjack, be documented.  That is done for the first time here.

The following is a moderately edited version of communications between me and the Dubner family in January, February, and March of 2017.  The first person singular refers to Dubner’s son, Robert.  Comments by his wife Harriet, who I will note had a career as a speech pathologist, are enclosed in square brackets.  Comments by me relating the content of phone and mail conversations with the family or elaborating upon or clarifying concepts are enclosed in curly brackets.  

Dubner’s Professional Career

He started working in 1949, just before graduating from college.  His first jobs out of college were with the Arma Corporation (1949–1951) and with the Avion Corporation (1951–1960).  Those jobs involved defense work, including fire control strategies for U.S. Navy surface vessels at Arma and the design of the infrared seeker head for the Navy’s Sidewinder missile, his main project at Avion. 

Professionally, in the 1950’s my father worked on missile guidance systems, in particular infrared homing systems. He was part of the team that developed the AIM-9 Sidewinder. He designed the optics, which meant ray tracing, which meant a number of people, mainly women, doing arduous calculations with mechanical calculators.  That led him to learning the capabilities of electronic computers.  He got hold of his first computer, a 1956 Royal-McBee LGP-30, in part to do those calculations. {LGP is the acronym for  “Librascope General Purpose” and, later, “Librascope General Precision.”}

He also became fascinated by catadioptric telescope design as a result of his optics work. In particular, he was much taken with the elegance of the Maksutov design.  In 1976, having been interested in astronomy and star-gazing all his life — when I was about nine years old he got me an Edmund Scientific three-inch Newtonian as a Christmas present, and he messed with it at least as much as I did — and having lusted after one for years, he bought a Questar Standard 3.5-inch telescope.  I have it now; I recently had it serviced and converted to the Duplex model, where it can be removed from its alt-azimuth mount and placed directly on a tripod for field work or as a telephoto lens. 

[We moved to Ridgewood, New Jersey, in 1963 and he worked at Simmonds {Precision Products}.  I believe he went to work at Curtiss-Wright in Paterson, New Jersey, in 1960 for a couple of years and then Simmonds also for a couple of years.  In late 1963 he went to work at Computer Applications until that company went bankrupt and then he formed his own company, Dubner Computer Systems.]  {Robert Dubner provides corrected dates.}  He went to Simmonds in early 1963, and from Simmonds to Computer Applications sometime around 1967.  Dubner Computer Systems was formed in mid-1970.

Although by the end of 1964 my father was at Simmonds, I am pretty sure that he was at Curtiss-Wright in the 1962–1963 time frame.  We moved from Washington Township, New Jersey, to Ridgewood in June of 1963.  Curtiss-Wright had many locations.  I seem to recall that my father worked at the Curtiss-Wright location in Paterson.  We used to go as a family to a restaurant named Steve’s which I believe was near where Dad worked.  A dim memory has something on the wall of the restaurant relating to the Kennedy assassination, which took place in November of 1963.

Curtiss-Wright was one of the country’s leading aerospace engineering companies.  The Paterson plant was a major one.  My father reported directly to the “big boss,” the plant manager.  My father’s title was “Chief of Advanced Design,” and at the time he was in charge of the biggest project going on there, a huge flight simulator.  There were many dozens of engineers involved: mechanical, electrical, production, test, you name it.  They all reported, directly or indirectly, to my father on that project.  As far as I know, this was the first major flight simulator based on a digital computer, rather than primarily based on analog computers. I regret to report I can’t recall what aircraft it was.  I don’t think I ever knew, although I can describe in detail how he and his team decided to use a digital computer at its heart.  My father can’t remember the aircraft either. 

I became interested in my father’s career and engineering in general when I was in the sixth grade, which was the 1964–1965 school year, my having been born in 1953.  I was eleven years old that Christmas, and I remember a class project involving a mural depicting a winter scene.  I augmented that mural by incorporating two NE1 neon bulbs as the eyes of the snowman. Powered by a 90-volt radio “B” battery, a simple RC driving circuit caused the snowman’s eyes to blink. 

(My teacher, who I realize in retrospect was a very sweet woman in her twenties, was utterly bewildered by me. I mean, c’mon.  I was a scrawny eleven-year-old kid, not even five feet tall, and I show up with a handful of electronic components and a 90-volt battery, connect them together with some alligator clip jumper cables — without a drawing — and the lights start blinking. She must’ve felt like she was falling down the rabbit hole.)

The key thing is that my dad brought those components home from his job at Simmonds Precision Products in Tarrytown, New York.  The other key thing is that I was, at around that age, starting to pay attention to my father’s work and career.  I relate the anecdote about the snowman’s eyes to establish my bona fides and to mark that point in time. 

A few years later when I was in the ninth grade, the 1967–1968 school year, I learned to program computers.  I worked with my father for the next thirty years, so I can lay some claim to being able to come up with some good guesses about what happened in the times before I really started paying attention. 

Curtiss-Wright still exists, as does Simmonds Precision Products.   It is an 85-year-old aviation company.  The “Wright” goes back to the Wright brothers and the “Curtiss” goes back to Glenn Curtiss.  Curtiss and the Wright brothers were vigorous competitors in the early days of aviation but eventually their companies merged.

Either Curtiss-Wright or Simmonds Precision Products would have had, in 1963, a print shop capable of creating those 3 x 5 cards. 

Early Interest in Blackjack

Starting around 1950 when he was twenty-two years old, the business and professional work at both Arma and Avion occasionally took him to the West Coast and he developed a habit of stopping overnight in Las Vegas on his way home to New York probably both for conferences and for convenience in travel.  Because traveling from California to New York by turboprop aircraft was noisy, lengthy, and tiring, it was much more civilized to take a short hop to Las Vegas, spend the night there, and then take an early flight back to the East Coast than it was to travel by airplane at night.  There, other engineers introduced him to Las Vegas and casino gambling.

Although he wasn’t a gambler, he loved games and puzzles.  He also loved the hustle and bustle and sheer weirdness of Las Vegas; we talked about that many times. He was a mathematician and an innovative thinker, and, at some point after learning to play blackjack, he figured out that unlike craps or roulette (which are pure games of chance) or poker (which is mainly a game of skill) blackjack offered the opportunity of informed advantage play.  It was during these visits to the casinos of Las Vegas that he developed his interest in possible advantage play in blackjack. 

It was obvious to him that knowing what cards had been played could lead to an advantage over the house, if only the task of recalling those cards and the resulting calculations could be made manageable.  But it was after he started using computers in the late 1950’s that he began thinking about using computers to develop a method that could be used by a gambler in a casino.

The Computer Used in the Blackjack Simulations

He easily could have started the {blackjack computer simulation} work at Curtiss-Wright and kept it up at Simmonds.

I am pretty sure the computer my father used at Curtiss-Wright was the IBM 1620.  The computer work was probably done off-hours at Curtiss-Wright.  [I agree.  It was at Curtiss Wright on off hours.]  My father had some big spools of black eight-track punched paper tape in his files. (I later worked with such tape for several years, so I have clear retrospective memories that it was the eight-track ASCII type, and not the earlier six-track Baudot coded tape used in Flexowriters.)  As I type this, I am having a visceral memory of how that heavy oily paper looked, felt, and smelled, as if were in my hand right now.  I remember my father telling me that those tapes were the saved output of blackjack simulation runs.  I am pretty sure the computer my father used at Curtiss-Wright was the IBM 1620.  I can’t swear to it, but the timing works.  And the 1620 had the 1621 paper tape reader and the 1624 tape punch. 

Although his memory is weak on this point, the work had to have been at Curtiss-Wright.  He might have done some preliminary work at Avion on the LGP-30, but the LGP-30 used six-level punched tape.  My memories are as clear as they can be after over fifty years, and although six-level and eight-level tape are each one inch in width, the hole patterns are different, and dang it all, the blackjack results were punched on the eight-level tape used on the IBM 1620.  And, again, there doesn’t seem to have been enough time after starting at Simmonds for him to have done the necessary non-work-related research there.

I know Harvey worked with the 1620. I have no memory of him working with the IBM 704 or 709. That doesn’t mean much because until I started programming in 1968 I didn’t pay attention to the “which computer” question.  I know his first machine was the Royal McBee LGP-30, and I know about the 1620.  But I don’t know much about other machines he used at work until I really got involved around 1971.  Online articles describing the IBM 704/709 don’t mention punched paper tape. Those machines apparently used magnetic tape along with card punchers and readers for data storage.   And I “know” he punched the output of his blackjack simulations on paper tape.  This all points to the 1620, and away from the 704/709. 

Without being sure, my guess is that he did the blackjack simulations on an IBM 1620; the timing is about right.  [We moved to Ridgewood in June 1963 soon after Harvey went to work for Simmonds.  But I think he did his simulations when he worked for Curtiss-Wright.] 

{Only mild uncertainty remains in the family memory whether the blackjack simulations were performed at Simmonds or Curtiss-Wright.  The family moved in June, 1963, after which he began to work for Simmonds, and the 1963 Las Vegas conference was held in November, 1963.  This probably allowed insufficient time to prepare the paper and submit it for presentation, implying the work was performed before the move, at Curtiss-Wright.}

I don’t believe he “worked at night” on the computer.  He would have written the code at night at home.  He would have gotten it working during the work day.  We always mixed in extracurricular stuff in with the paying work.  It was fun, it kept things interesting and exciting, you learned things that way, and the paying work “always” got top priority, with the extra stuff filling in the cracks.  Then he would do the long runs for statistically significant results overnight, gathering the results the next morning.  {Because of the relative slowness of computer processing, this was the mode of computer analysis of time-consuming programs in the early days of computing.}

A “long run” for him, given the power of the computers of the time, was on the order of ten thousand hands. 

The output paper tapes from his computer runs are long gone, along with boxes of IBM cards, 8-inch and 5.25-inch and 3.5 inch floppy disks, and piles of green-striped computer printer paper.  When I talk about seeing the paper tape, I remember seeing it in the house we lived in before we moved to Ridgewood in 1963.  I was ten years old when we moved.  I used to play with those tapes when he was done with them.  By the late 1960s he wouldn’t have had access to an IBM 1620 and the tape would have been so much scrap; they would have been discarded by then.

The sad thing is, he almost certainly had notebooks from that time, and he likely would have recorded his work in them, too.  I had a look around the house a few weeks ago, but the earliest of his notebooks seem to be from around 1980 or so, which is when we started the real push into large-precision arithmetical number theory.  The older notebooks don’t seem to be around.  Back around 2006 or so my parents had a flood in their basement, and it’s very possible those earlier notebooks were damaged beyond recovery at that time. 

Dubner’s Approach to the Blackjack Problem

As to what happened during the development of his counting rules, my knowledge of my father and the way he liked to work tells me it went like this.  He played blackjack from time to time in Las Vegas and got interested in the possibility of advantage play.  He would not have attempted to find anybody else’s work on the subject.  He and I talked about blackjack a lot over the years, and he never mentioned anybody but Thorp, but he never indicated that he learned about counting from Thorp. 

Having gotten interested, he would have done a mathematical analysis.  Taking the formal math as far as he could, he would then have started writing computer code.  He would have investigated random number generators and created his own.  He would have figured out how to use an RNG to shuffle a simulated deck of cards — which you may know isn’t actually as straightforward as it sounds; there are a lot of ways of screwing up both random number generation and shuffling — and then he would have started playing simulated games. 

I don’t know the details of blackjack, but I believe there are some straightforward rules for when to ask for additional cards based on your hand and the dealer’s up cards. And once you have rules of play, you can simulate multitudes of hands and experiment with methods of counting and calculating your edge. 

To elaborate on this concept, in one of your questions, you wonder if “intuition” had something to do with it.  I can assure you that intuition was involved in the process of developing the hard theory, but it was the hard theory and the results of simulations that led to his results.  This is how mathematical and scientific research works: an investigator starts with a suspicion, an intuition, a hunch, maybe even a wish.  But those feelings have to survive rigorous analysis.  Otherwise they are just guesses.

I speak as one familiar with this kind of thing. I know how I would respond if I were to get interested the way I believe that my father got interested.

First, I would learn how blackjack is played.  I would have learned the rules of the game, and I would have investigated optimal play in the case where I have no knowledge about the constitution of the deck.  I am talking about things like, if I am holding two tens, it’s kind of stupid to draw.  And if I am holding a ten and a two, it’s kind of stupid to stand.  (At, least, I think that’s the case; I am not a blackjack player.)

And I would have verified mathematically that when the deck is rich in high cards that I am more likely to win, and when the deck is rich in low cards I am more likely to lose. 

Operating on that omniscient level I doubt any simulation is necessary; if you know exactly what cards are left in the deck, then the probability of winning or losing can be calculated precisely. 

I believe that for a full deck, this has long been worked out and represents the “basic strategy,” which is how to best play your hand against each possible dealer upcard.  That procedure is also the foundation behind casino rules and payouts; it’s what makes blackjack the game it is.  But all of those conclusions are based on averages; they would have assumed that you are playing against an infinite deck of cards.

Knowing my father, he would have built a blackjack engine and run it some thousands of times in order to verify the “basic strategy.”  He would have done it for two reasons:  First, to verify the published basic strategy.  Second, in order to test his blackjack engine.  If for any reason his results differed from the published versions, he would know that he had to check his implementation of the random number generator and his “basic strategy” decision making.  He wouldn’t expect to discover the published techniques to be wrong, but you never know.  And it’s very easy to make implementation mistakes, so he wouldn’t have moved forward on a belief that the published techniques were incorrect until he had really verified everything else.  The basic principle is that it’s necessary to make sure your initial conditions match what came before you.

Now we get into “informed play”:  If you know the composition of the remaining cards in the deck because you kept track of every card that was played, you can, in theory, calculate precisely the odds of winning your next hand.  It’s a combination, of course, of every possible hand the dealer can deal both you and himself, but it is in principle calculable.

That’s where artistry and simulation come into play.  Perfect knowledge is too costly for a human to work with; he wanted to come up with something simpler, something that could be done without equipment at a casino table.  He would have started with various scenarios:  What if I know exactly how many aces there are left, and how many face cards, and how many twos, threes and fours?  Something like that.  Then he would have started simulating:  The computer would play thousands of games.  He would have grouped each hand into buckets based on the strategy he was testing at that time.  He would have kept track of how many times each bucket resulted in a win.  I feel confident that he iterated through that kind of process until settling on “count the high cards, count the low cards, subtract the lows from the highs, and divide by the number of cards left.”

So the process would have started with the conviction that an edge was possible even with “chunky” knowledge of the composition of the remaining deck, rather than precise knowledge.   Informed intuition would have led to the creation of various scenarios.  Simulation would allow each scenario to be tested, and the results of the tests would lead to one that seemed optimal:  That is, it got “good enough” results without requiring superhuman memory or arithmetical abilities.

What I don’t know is precisely what he was testing in those simulations.  I do know the results he arrived at.  And I worked with him often enough on other similar investigations to be morally convinced that he arrived at the hi-lo index by testing a number of other, similar, ways of counting the cards, and keeping track of which method seemed to work best in terms of being feasible at a casino table and effective in generating returns.

You ask about the term “basic” on the Hi-Lo card.  I have no direct knowledge of why he used it.  But knowing my father, I am positive the word means, “There is more to the story.”  As I mentioned earlier, the method on the card wasn’t the best possible method.  It was my father’s attempt to balance practicality with effectiveness. 

Independent Derivation of Optimum Bet Size Criterion

It was at that point that Harvey ran up against the question of given that you have an edge, how much do you bet?  Too little, and your winnings grow slower than they need to.  Too much, and you’ll go broke if there is a run of hands against you.      

The simulations would have shown: “When the hi-lo index is “this,” then the probability of winning is “that.”  The next question becomes:  “If I am playing blackjack and my probability of winning the next hand is 52%, how much should I bet?”

Imagine a coin flipping game:  Tails means you win your bet, heads means the casino takes your money.  You somehow know the coin is biased to come up tails 52% of the time, and you know that if you play long enough you’ll come out ahead.  How much should you bet?  If you have $100 and place it all on the table, you have a 48% chance of going broke on the first flip.  Bet nothing and you might as well have stayed home. 

You can infer that there is some optimal amount to bet to maximize the rate at which you make money, given the magnitude of the edge and the amount of money in your wallet.  That’s what my father figured out, and he did the analysis, and he came up with the optimal amount to bet once the card counting rules gave him the size of the edge. 

So, given a known edge in the game, there clearly is some point between betting nothing and betting everything you have that will maximize your winnings per hand over many hands. That can be plotted and analyzed as winnings-per-hand against size-of-bet (with both axes expressed as a fraction of your stake); it is an upwardly convex curve {imagine an opened umbrella held overhead} with the optimum at a peak.

That point can be analytically found using simple calculus.  Finding good numerical values would be trivial using computer simulation.  I asked my dad about it once, and he told me that he had done the math. 

This is exactly what John Kelly did and published in 1956 and which became known as the Kelly criterion, or a Kelly bet.  He learned sometime later that he had independently recapitulated the work of John Kelly.  I am making educated inferences about some of this, but he told me he learned of the Kelly criterion after he had recreated it himself.  {He told me as early as 2006} that he had rederived the Kelly criterion and only later learned of Kelly’s work. 

So, there it is.  Some of that I know my father did because he told me:  He independently duplicated Kelly’s math, and I know he simulated tens of thousands of blackjack hands.

Family Claim that Dubner Was Not Aware of Thorp’s Early Work

Based on my recent long conversation with my parents, I hadn’t fully realized how much casino blackjack he’d been playing during the 1950s.  It’s my understanding that has grown over the last few weeks.  I hadn’t realized just how much blackjack he played in Las Vegas during the 1950s. 

My father, through the years, and during recent discussions inspired by your questions where we have walked through his memories, has always said that he never even knew who Thorp was until the 1963 conference.  We’ve been talking to him about Thorp, and he says definitely that he didn’t know about Thorp’s work prior to 1963.  He and my mother are in agreement that he played blackjack in Las Vegas starting as early as 1950 or so, and that his interest in advantage play grew steadily throughout that decade, culminating in his developing his method through computer simulation after the computers became available.  My belief now is that the work was being done in parallel by Thorp and my father; neither knew about the other.

{The independence of effort in developing the Hi-Lo strategy is confirmed by blackjack writer Peter Ruchman, who quoted Thorp as writing:  “At the 1963 conference Harvey Dubner presented the . . . (high-low system), which he had thought up himself.}

As I noted, his work taking him to Las Vegas in the early 1950s led him to start playing blackjack.  Being mathematically inclined, he would have quickly figured out the theoretical possibility of advantage play.  After he started using computers in 1956, he would have automatically started thinking about using them to develop workable techniques.  He remembers doing just that.

One thing my father is adamant about: He worked out the possibilities of what became his hi-lo strategy on his own.  It grew out of playing casino blackjack from time to time during the 1950s and the early 1960s.  I have repeatedly revisited with him the question of “Where did you get the idea that counting the cards could lead to an edge over the house?”  His answer is unwavering: “I worked it out myself.”  When asked if he learned of the possibility that he learned of it from Edward Thorp, his unwavering response: “I never even heard of Thorp.”

{That his work was at least begun without knowledge of Thorp’s 1962 book} makes sense.  There just wasn’t enough time between publication of Thorp’s book in November, 1962, and the November, 1963, conference for my father to have developed the expertise to play casino blackjack, work out the math, write the programs, do the simulations, and develop what he referred to as “his method.”  Between Beat the Dealer coming out in the fall {November} of 1962 and my father starting a new job in 1963 there just wasn’t enough time to do the work the way he did it.  He had to have been working on it before Thorp published. 

He couldn’t possibly have learned about counting from Thorp’s book in 1962, then become competent at casino play, figured out the math, worked out the concept, written the code, and developed his method all between the book coming out and then starting a new job in early 1963.  If nothing else, he wouldn’t have been in a casino often enough.  And this was all extracurricular work.  He wouldn’t have let it interfere with his job, so his time commitment wouldn’t have been great.  I had previously thought that he learned about it from Thorp but that can’t be right; there just isn’t enough room in the timeline.  And when I was doing that speculation, I didn’t know he’d been regularly going to Las Vegas during the 1950s.

I figure also that card counting research and development in the early 1960s was a case of “when it’s time to railroad, railroads pop up all over.”  I’ve been told that various gamblers had been trying to count cards for years.  They kept quiet about it, so as not to lose what little edge they had, but word was spreading.  J.L. Kelly published his work on optimal betting in 1956.  Computers were becoming more generally available.  So it’s not surprising to me that a lot of activity was happening in parallel. 

[Harvey says he was not working with anybody when he created his Hi-Lo system.  And no he did not have a software collaborator.  He was not in contact with Julian Braun.  He says he might have been in contact with him after the {1963 Las Vegas computer} conference, but not for any reason.  Harvey is sure he never heard of Thorp until the 1963 joint computer conference.  He is sure he worked independently.  Harvey said he created the {3 x 5} card before he even knew who Thorp was.]

So, putting together everything I know now, it’s clear to me that Harvey was one of a small number of people at that time who had the interest, the inclination, the mathematical training, the innovative insight, the computer programming ability, and — not insignificantly in that time frame — the access to a computer necessary to conceive and build the simulations needed to create and test useable blackjack counting strategies.  But it is clear to me that he started his work because of the possibilities that became apparent to him while playing and thinking about blackjack, and not because he learned about card counting from anybody else.

Concerning the comparison of the Hi-Lo and Ten-Count strategies {displayed in the apparent slide}, does it demonstrate that Harvey knew about Thorp and the Ten-Count Strategy at the time of the conference?  Sure does.  That leaves the question you find most interesting:  Does it show that Harvey learned about counting from Thorp?  Harvey is adamant that he developed the idea on his own.  My belief, knowing him, is that he did develop it on his own.  My belief is that when he learned of the Ten-Count strategy he compared it to his method, found his to be superior, and that may well have led to him deciding to present the Hi-Lo strategy at the Fall 1963 conference.  {Or, he may have already decided to present at the conference before reading Beat the Dealer and simply added the comparison to the material he had already planned on presenting.}

Once he learned of the ten-count strategy he would very naturally have compared it to his.  And upon finding his Hi-Lo strategy to be superior, and then learning of the panel session at the Fall Joint Computer Conference, it’s natural that he would have used it as an opportunity to go public.  It’s distinctly possible that he would have gone to that conference even without the “games of chance” session; he was a computer professional at a time when it was still a new field. I note that he had a paper published at the 1970 Spring Joint Computer Conference  (http://dl.acm.org/citation.cfm?id=1476964).  I am speculating that he was planning on that trip to Las Vegas anyway, and the panel discussion was a chance to have some fun and show off his work.

It seems clear that by mid-1963 my father had read the 1962 Beat the Dealer, because it appears from that handout {Figure 3} that he tested Thorp’s ten-count system using the same simulation techniques he used for developing his Hi-Lo.  {The format employed by Dubner in the handout does not allow immediate comparison with Thorp’s published results, and this comparison would likely have required the additional simulation to which Robert Dubner refers.}  But there is no question in my mind that he did the bulk of his work before he read Beat the Dealer.  He started his work and he developed his methods, and then read Beat the Dealer.  Remember, he wasn’t a professional gambler.  His interest was theoretical and not because he wanted to win at blackjack.   He wouldn’t have bought the book except for the fact that he was already doing research and development on card counting.

How much of his final work was done after reading the book versus before nobody will ever know.  My guess is “not much”; the essence of my dad’s method is that it is simple; it would have coalesced fairly early. 

{The 1962 copy in their family library is the hardcover first edition from Blaisdell Publishing Company.}  I see one rare book dealer offering one in fine condition for $500.  My mother later commented that maybe we should sell it. 

{The findings of the family of the copies of apparent slides which may also have been used as handouts at the 1963 conference led to a further consideration.}  There are a number of interesting things in that handout.  He apparently did some simulations that involved hundreds of thousands, even millions of hands.  Pure speculation, of the type that I really should stop making:  I know he has told me in the past that his runs involved tens of thousands of hands. This leads me to wonder if he started his work on the slower Librascope LGP-30 {while he worked at Avion, from 1951–1960}, and later refined it on the faster IBM 1620 {at Curtiss-Wright}.  But that’s absolutely pure speculation.  {If true, Dubner clearly began his investigations before the publication of the 1962 first edition of Beat the Dealer.}  The letter-sized sheet of paper was prepared at the same time as the 3×5 card, which means at some point in late 1962 or early 1963 when he still had access to the 1620 computers at Curtiss-Wright.

If Thorp’s account in the 1966 edition of Beat the Dealer is accurate, Harvey didn’t present those sheets at the meeting.  Thorp says on page 94 of the paperback 1966 edition {of Beat the Dealer} that “Exactly how much better or worse (the Hi-Lo system) is than the Ten-count method is not known.”  It sure doesn’t sound like Thorp saw those sheets of paper.  My dad was a natural showman; he may have decided that the sheets were too technical and he should just stick with the “We’re in Las Vegas, and with this little card you can beat the house!” fireworks, and leave presenting the math for another time…a time that never came.  {I’d be shocked if Harvey didn’t present the sheets, as handouts or slides/handouts.  This was a technical meeting and he wouldn’t just provide the slide of the 3 x 5 card. Thorp, as stated, remembers that the material was presented.  His comment in Beat the Dealer may resulted from either not at the time of its writing remembering the details of Dubner’s presentation or not wishing to quote results from a study the details of which he had not examined or verified for himself.}

I can’t comment on any possible competition in the development of the theory of card counting, or who might have appropriated work from somebody else.  I am certain Harvey did his work independently.   My understanding is that Mr. Thorp, in the later edition of Beat the Dealer, acknowledged Harvey’s work and credited him with putting card counting onto a solid mathematical footing.

I very strongly doubt that he even knew of anybody else’s work, much less was taking anybody else’s work into account.  I don’t know if he had published any papers at that point in his career, but he shortly would start to do so.  If anybody else’s work had informed his, I am certain he would have credited them.  And I’m pretty sure it would have come up in our conversations over the years. 

For complex problems, especially work related, sure, he would research, and collaborate, and build on the work of others.  He did that a lot.  But he was doing the blackjack thing for fun.  He paid a lot of attention to blackjack over the years, and many of his memories are pretty clear. He doesn’t remember where he did the work, but he does remember doing it, and he remembers why he did, which was for the fun of it and to prove that you could get an edge over the house in casino blackjack. 

He wouldn’t have found it to be fun to start with somebody else’s work.  It would have been far more characteristic of him to start from scratch.  He was not surprised to learn that he was among a group of people doing similar work.  To some extent it was the availability of computers that led both Thorp and my father to do the work they did.  When it comes time to railroad, railroads pop up all over the place.

In addition to his current assertions, I had many conversations with dad about blackjack and counting over the years.  He never mentioned Thorp except in the context of what happened after the 1963 conference.  He never once said that he learned about the possibility of getting an edge from anybody else.  He always spoke of it in terms of figuring out that it could be done and working out how on his own.  He never spoke about building on anybody else’s work, or working with anybody.  He always spoke about working out the possibility himself, and working the math and the computer simulations to come up with his method.   He never referred to it as anything but “his method.”  For example, he would talk about how “theoretically there are better methods than ‘my method,’ but they are much harder to use, and the gains are small.”  He didn’t even use the name “Hi-Lo” when talking to me, and he didn’t even call it “card counting” much, except when talking to other people.  When he and I discussed it, it was always just, “my method.”

I have always been interested in the technical aspects of his work on blackjack, and we talked about it many, many times.  He always — always! — spoke in terms of what he had figured out.  He never once said that he had learned about the possibilities of card counting from somebody else. 

And if he had, he would have acknowledged it.  I worked with him for years doing number theory research.  He learned a lot from other investigators, like Richard Brent, and Peter Montgomery, and Hugh Williams, to name just a few.  He often spoke in terms of what he had learned from those other people.  He was always careful about what he learned from other people, and the original work he did himself.

If the idea that Harvey did the work all on his own is controversial, well, it’s only because nobody ever asked him.  He’s been in the phone book all this time, and for many years his e-mail address has the same and available.  If anybody asked, as you are asking, how, he would have said, as he is saying now, that he did it all on his own. 

If somebody can demonstrate that they were in touch with my father about blackjack before he gave that presentation, I would be both astonished and incredibly interested in talking to them. 

 If somebody thinks they can prove my father read Thorp’s book {before he began his own investigations}, or somehow got the idea for card counting other than on his own, well, I’m pretty sure that his reaction over the years would have been to ignore them.  He was always like that.  He knew what he’d done and, although he liked being acknowledged for it, he didn’t go out of his way to correct misapprehensions.  Heck, I happened to pick up a copy of {William Poundstone’s} Fortune’s Formula a few days ago.  Lo and behold, my father’s counting method is described in there, but without attribution, except vaguely in Thorp’s direction.

So it’s not surprising that dad’s method evolved into being described as “an improvement over Thorp’s method.” 

It’s incontrovertible that Thorp did publish before my father gave his presentation in 1963, so it’s perfectly reasonable for the world to have concluded that Dubner’s method was an improvement over Thorp’s.  But Harvey is clear that he didn’t know about Thorp’s work, and his doing the work de novo fits every discussion he and I ever had about it.  {At this point, after discovery of the apparent slide comparing the Hi-Lo strategy to Thorp’s ten-count strategy, Robert Dubner would state that his father did not know about Thorp’s work at least when he began his investigations and possibly until most of it was completed.}

Presentation of Hi-Lo at the 1963 Fall Joint Computer Conference

According to the program as published in the {September, 1963, issue of the} trade journal Electrical Engineering the conference was to take place in the Las Vegas Convention Center.  You’ll note at the bottom right of page 20A in the program {see Bibliography} that on Wednesday, November 13, at 8:00 p.m., “Computers Applied to Games of Skill and Chance” was scheduled, chaired, inexplicably, by “R. A. Kudlich, General Motors Corp.” 

In the “backmatter” {of the published proceedings; see Bibliography}, “E. O. Thorp” is listed as one of the Session Chairmen.  {This information, however, was published in the proceedings of the conference, and not before it.  Dubner first encountered Thorp at this conference, according to his recollection.}  In the second big box on the page {see Bibliography} there is a set of tabs, labeled Abstract, Source Materials, Authors . . ..  The Source Materials tab has a link to the Back Matter. 

{The proceedings were published both by the Association of Computing Machinery and Spartan Books of Baltimore, Maryland.  Neither, however, provides excerpts from the games of skill and chance panel discussion, only papers formally presented to the various conference sessions.  The 3 x 5 index card on which Dubner printed the “Basic HI-LO Strategy” and, if they were presented, the nine slide/handout material, are the few, if only, artifacts remaining from the panel discussion on games of skill and chance.}

Dr. Kudlich was a long-standing and reliable administrator of the IEEE, and he apparently organized the 1963 “Computers in Games…” panel.  It seems equally clear to me that Thorp was brought in as the pre-eminent popular figure, because of Beat the Dealer, to chair the actual session.  (An inspired move, if you ask me!)  My educated guess is that Kudlich was probably there, probably introduced Thorp to the enthusiastic crowd, and then Thorp moderated the discussion.  In short, it really looks to me like Thorp was rightly brought in as a celebrity figure to moderate the panel.  {At the date of this writing, Thorp has not been able to provide information from his files as to when he was invited to moderate the panel, when and how that development was publicized, if he selected the members of the panel, and if Kudlich in fact introduced him.}

I asked him about how he got to present at the 1963 conference.  He said that he learned about the panel discussion on computers and games of chance — the topics are published in advance — and he contacted the conference and asked for a slot.  {That is, his participation was not based on personally knowing Thorp.}   (My father is a lifetime member of the IEEE, and Electrical Engineering Magazine was an organ of the IEEE.)  And he simply doesn’t remember.  The answer {to exactly how he learned of the panel discussion}, again, is that nobody knows; nobody can know.   But you can be absolutely sure that Curtiss-Wright had a library that subscribed to all the publications of all those groups, as well as the publications of the American Federation of Information Processing Societies.  At that time a company like Curtiss-Wright would have had all of them available, and my father and his coworkers would have paid attention to them.  The astonishing thing would have been if a panel on “Computers Applied to Games of Skill and Chance” had gone unnoticed by my father, given his interest at that time in blackjack.  I don’t know who R. A. Kudlich was, but it’s easy to think that when that September 1963 magazine was put to bed Edward Thorp hadn’t yet agreed to chair the panel. 

Casino Reaction, Dubner’s Personal Blackjack Winnings, and His Personal Reflections

I know people who have made tens of thousands of dollars using his method.  Hell, I know a guy who claimed he bought a house with card-counting winnings.  But not my father.  As my mother and I have said, he wasn’t a gambler.  He is, in fact, remarkably risk averse.  He has told me that when playing at a one dollar table when the deck got hot, his hand would shake when he put down the twenty dollar bet. 

The pickings were easy in the beginning.  The casinos didn’t know what was going on, and particularly downtown in Vegas the dealers would play a single or double deck right down to the bottom. 

As time went on, of course, you know what happened.  Two, four, even six decks.  The yellow “stop here and shuffle” card halfway down.  Lately the casinos, I am told, have deployed their Doomsday Machine: continuous mechanical shuffling, wiping out any hope of an edge. 

And, of course, the dealers themselves had nothing better to do than count the cards, and when they saw the counting betting pattern they’d signal the pit boss, who’d intervene. That led to the team play concept made noisily public by the MIT crew, where one person would quietly count and signal a cowboy to come play a few hands when the deck got hot.  But you know all about that. 

People have a tendency to gleefully assume that my father was banned from casinos all over town.  Not so.  Nobody knew who he was by sight.  He kept his head low when he was playing blackjack.  The one and only time my father was asked to leave — it was at a downtown casino, I think he once told me — he was actually behind at the time.  “But I’m losing!” he told me he told the pit boss.  The polite reply was something like, “We don’t want your money.”

[He never stayed long at any casino.  He didn’t want them to know he was counting.  Even so, at one point we were asked to leave although he was betting very little.  I don’t remember the name of the casino.  We stayed at the Riviera so I guess we played more there than anywhere else.  We also played in town.]

My father wasn’t there to win.  He was there for the satisfaction and fun of experiencing his results at work.  He kept careful records; I am sure you will start slavering when I tell you those records still exist, as far as I know.  Among other things there is a graph, covering at least a decade or two of once or twice a year visits to Las Vegas, showing the amount of money earned per hour of play.  It slopes steadily down to the right, both because of some outsized returns early on — a case of “beginner’s luck” — combined with the effects of the steadily escalating countermeasures by the casinos over the years. 

[He says what started out as an interesting mathematical problem became a big thing.  And as time went on, it became bigger and bigger.  At the beginning, he had no idea of the ramifications of his discovery.  Harvey says suddenly he was involved in a situation where he was making money and had a good chance of making a lot of money some time in the future.  But he was doing other things professionally that were even more interesting.  He was doing more and more work that was much more significant than card counting.  Remember always he was a mathematician, not a gambler.  He never benefited financially from his blackjack creations. He was a mathematician, never a gambler.] 

I will reiterate the point:  My father wasn’t a gambler, and he had no particular interest in blackjack.  But having learned that it might be possible to get an edge on the casino by tracking what cards had been played and thus developing some knowledge of what was left in the deck, and having his interest piqued by that, he would have —  based on many discussions about this over the years — done the analysis, done the math, written the code, done the simulations, and formulated his method, completely on his own.       

He was proud of how easy his method was to use.  Perfect counting could do a little better than his rules, but it required perfect memory and a complex series of responses, and it could only do a few percent better.  At least, that’s what he told me. 

So he twisted the tail of the casinos with that 1963 paper.  I am absolutely sure he did it on purpose; his sense of humor is extremely puckish.

{He did make enough money on one trip to buy his wife a $200 diamond pin, which they still possess.}  [We bought the pin in Ridgewood, New Jersey.  I believe the name was Webber Jewelers.  That’s just a vague memory.]  The jeweler in Ridgewood was Weber Jewelers.

You wonder if those sheets of paper {containing the nine materials} were created after 1963 for a second presentation.  I tell you that they were not.  There is no evidence anywhere that my father ever talked publicly about blackjack counting ever again.  He talked about it plenty privately.  He made no secret about it.  He relished having created the first practical blackjack counting system, and would tell anybody about it at the drop of a hat.  But he wasn’t part of the gambling world, and didn’t expend any energy pursuing it.  He never did any subsequent research or simulations. 

After 1963 he was done, except to go to Las Vegas and occasionally Atlantic City and count at blackjack, keeping track of his winnings.  His long-maintained gently downward sloping graph of $/hour over time appears to be lost, but he was still winning when, around 1977, not long after the sole time he was asked to leave a casino because he was caught counting, he gave it up.  I asked why he stopped.  He said it was because it had become boring.

But my father had a unique opportunity.  A mathematician at heart, he liked to prove things formally.  As an engineer, he would switch to pragmatic methods when the formal math fell short.  Having embraced computers for solving problems starting in 1956, he knew when to switch from formal methods to computational ones.  He had the computer programming skills to implement the methods, and he had access to the computer needed to perform the calculations.  He also had the wry sense of humor needed to publish his results in Las Vegas.

Encyclopedia Reference: Harvey Dubner
More Information: Hi-Lo Strategy
Book Review: Leslie M. Golden
Buy This Book: Never Split Tens


Full-Time Gambling — Viable Profession or a Fool’s Trap?

Professional GamblingBy Trent Sissoko

If you’ve ever had a couple of successful gambling rounds, you’ve probably wondered whether the thought of becoming a full-time gambler is plausible. Even if it is, is that a practical career choice? Some of the most prosperous full-time gamblers would say that this career choice is definitely an option.

Take, for example, Dan Bilzerian and Matthew Benham. Aside from being an actor and a businessman, Bilzerian is a full-time gambler, while Benham is also the owner of Midtjylland FC and Brentford FC1)Midtjylland FC and Brentford FC are professional football (soccer) clubs.. So, why shouldn’t this career be an option for you.

However, even before you set your sights on expanding your gambling career, you should consider both the pros and cons of that venture. Keep reading to find out both the negatives and the positives of full-time gambling, as well as how to make it in this line of work.

Disadvantages of a Full-Time Gambling Career

Although there are plenty of benefits of being a professional high-roller, take a look at a few cons first.

  • High chances of failure — Before anything, there is a higher chance of failure than success with professional gambling. Compared to other occupations, full-time gambling is the one that could bring the largest financial crisis. Many failed gamblers have never recovered after a futile endeavor.
  • Vices — Because of the nature of the business, being exposed to vices is inevitable. If you fail to discipline yourself as a career gambler, you might wind up surrounded with bad choices.
  • Social stigma — The social stigma of being a professional gambler is still present. You might find yourself in a bit of a problem if you’re at a business or social event and you need to introduce yourself as a full-time gambler. Although people will find your career choice interesting, they’ll also probably give you side-eye.
  • Internal conflict — If you come from a religious background, you might find yourself in disagreement with your new way of life. Even if your gambling career turns out to be a success, the chances you’ll be struggling with internal issues are high.
  • Dependence — If this turns out to be your profession, you will be dependent on the whims of many gambling institutions. Whether you’re an offline or online gambler, you will need to keep up with the constant changes that could hinder your success.

Advantages of a Full-Time Gambling Career

Now that you’ve gained insight into the negatives of a professional gambling career, let’s look at the positives.

  • More time — With a career such as this one, you’ll have more time on your hands to spend traveling, with family and friends, exploring other options, etc. Professional gamblers don’t have 9-to-5 jobs, so their schedules are much more flexible.
  • Mental strength — Your mental capabilities will improve without a doubt. This line of work forces you to develop and practice your mental resilience that can help you in many other spheres of life.
  • New connections — Due to the nature of the business, you will be able to meet many new people and make new connections. This could pay off if you ever decide to venture out outside of gambling.
  • Celebrity lifestyle — If this is something you’re interested in, you could definitely become a celebrity. Thanks to social media, you will be able to live a celebrity lifestyle if you expose yourself enough. You probably won’t be an A-lister, but you will be more famous than you are now.
  • Travel — If you get good enough, you will be able to travel the world to play in first-class tournaments. That’s a win-win combination as you will be able to travel and earn money doing the thing you love.
  • Financial security — Although financial failure is an option, lifetime financial security is an option as well. If you manage to develop top-of-the-line playing skills, you’ll be able to secure your family’s financial security for many years to come.

Is It Worth It?

Ultimately, it’s all about finding the right balance between risk and reward. One of the most crucial things is to keep your head clear and make good and conscious decisions. Make sure you understand the odds of becoming a professional gambler, practice your discipline, and be clear about the advantages and disadvantages of this full-time position. Additionally, there are other things you should be aware of, so keep on reading.

Not All About Luck

People watching from the sidelines might think that professional gambling is all about luck. On the contrary — full-time gambling is anything but pure luck. Some of the best career gamblers are among the top strategists, which significantly increases their chances of succeeding.

That’s one of the reasons why you’ll see so many professional gamblers with the so-called poker faces. They know they will be able to make their way back to the top even if they have a bad hand. By simply sticking to a strategy, believing in yourself, and managing your bankroll, you will be able to achieve a lot and stay in control.

Embrace High-Skilled Games

Undoubtedly, playing online casino games or bingo is a great way to get you started. However, such games are not enough if you want to make a career out of gambling. With that in mind, sports betting and poker are two of your best bets to make it big. For that reason, the majority of players that want to become professional gamblers focus on poker.

There are plenty of advantages to playing poker once you upgrade your skills. First of all, you’re playing against other players as compared to the casino. That’s a positive thing as your chances of winning depend on your opponents’ skills, and the casino doesn’t have a say.

Secondly, once you get into it, you’ll be able to choose who you’ll play against. You will have a better understanding of the game and you’ll get to know the people you’re playing with. So, if the table you want to join has potentially strong players, you can move on to the next and keep your edge.

Bear this advice in mind as it will definitely help you if you decide to try out this profession and a great place to try get it started is at a €10 bonus casino.

Professional Gambling Is Hard Work

Contrary to popular belief, professional gambling is not a piece of cake. Although making large sums of money is probable, it takes a lot of hard work to get there. Before anything, you will need to be better than your opponents, and many of them have been doing this for much longer than you have.

Next, you will need to learn how to minimize your losses and maximize your gains when playing against those same players. To do that, you will need plenty of practice. Practice will help you build the right level of skill needed to venture into professional gambling.

So, before you decide to quit your day job, spend months practicing and bettering your skills. Once you see that you’re making continued success with your gambling, think about going full-time.

Have a Plan

Just like you would have a plan for any other business venture, you need to have a plan if you decide on a gambling career. This goes for every gambling session, not only the career in general. So, before you start your poker or blackjack tournament, make sure to have a plan. Think about when it’s the right time to give up, when you should stop reinvesting and take your earnings home, and similar. Ask yourself these questions every time you decide to gamble.

Revenge Gambling Equals Loss

Even though you might be disappointed or angry, revenge gambling is not the solution. There are countless stories of revenge gambling leading to tremendous losses and disastrous results. If you decide on revenge gambling, that means you’ll be leaving all of your strategy and hard work behind. Similarly, becoming a millionaire overnight does not happen if you lead your gambling career with revenge in mind.

Stick to Your Comfort Zone

Even before you start your gambling career, you will notice that you excel at certain areas more than others. If that’s the case, the best option is to stick with what you know. If you are good at poker, avoid the roulette table. If sports betting on cricket is your thing, do not waste your hard-earned money on basketball wagering.

If you have two or three areas that best showcase your gambling knowledge, don’t get greedy and spread out to numerous other areas. The point is that your success will be much more evident if you stay within your comfort zone.

On the other hand, if you decide to try other gambling activities, you should do so without investing real money. Start with free online games and practice your strategy. If you see that you’re doing well and that there might be a future for you in a particular area, widen your horizons. You can always update your portfolio with new skills; you just need to be patient and take things slowly.

Stay Grounded

One of the most crucial things in professional gambling is the ability to stay grounded. Winning thousands of dollars will easily get you over excited and ready for a new adventure, but you should lead with your head and not your heart. Having a lot of money at once might lead to making some bad decisions, like putting a down payment for a new house.

This also applies to your state of mind during the game. Try to keep your head cool and stay grounded at all times so as not to make any rash decisions that might lead to your downfall. Try not to put any pressure on yourself during the game as it can affect your success.

Final Thoughts

Now that you have a list of pros and cons of professional gambling, as well as some advice on how to make it in the business, you should decide whether it’s worth it. Although it is not a stable career, it can lead to opportunities and financial gain. If you’re serious about making gambling your career, try to be sensible, practice every day, and come up with strategies that will give you an edge. Good luck!



1Midtjylland FC and Brentford FC are professional football (soccer) clubs.

The Best Blackjack Books of All Time

By Michael Dalton

There are many good books that have been written about the game of blackjack. Here are my personal favorites for the best blackjack books of all time. These books have stood the test of time and provide outstanding advice on how to improve your blackjack skills. As a collection, these books cover everything you need to be successful at beating the casinos. Books are listed in order of the year they were first published. If you buy any of these books, be sure you get the latest edition. 1) Michael Dalton is author of the Encyclopedia of Blackjack and editor/publisher of the Blackjack Review Network. Looking for other good blackjack books? Check out the product review section of the Encyclopedia of Blackjack and the online book catalog on this site.

Beat the Dealer by Edward ThorpBlackjack Hall of FameEdward Thorp’s Beat the Dealer (First published 1962)

Why? This is the classic that changed the way we all view the game of twenty-one. This book presents, for the first time, a validated winning strategy (a ten-count) based on the results of computer simulation. The 1966 version has a practical point count (Hi-Lo) that was later revised by Julian Braun. Is this required reading for the aspiring card counter?  Probably not…. but if you are a history buff, it is a must read. Edward Thorp.

[ BUY ON AMAZON: Edward O. Thorp: Beat the Dealer ]

Playing Blackjack as a Business by Lawrence Revere Blackjack Hall of FameLawrence Revere’s Playing Blackjack as a Business (First published 1969)

Why? The classic text that many early card counters were taught by including myself. Contained nice color charts that made it easier to memorize basic strategy. Included four counting systems developed by Lawrence Revere with computer simulation data from Julian Braun.

[ BUY ON AMAZON: Playing Blackjack As A Business ]

Professional Blackjack by Stanford WongBlackjack Hall of FameStanford Wong’s Professional Blackjack (First published 1975)

Why? The best book to learn the very popular Hi-Lo card counting system. Be sure to get the 2011 (or later) version for updates and corrections to the count variation charts.  The Hi-Lo count is the most recommended count of all time. Stanford Wong.

[ BUY ON AMAZON: Professional Blackjack ]

Theory of Blackjack by Peter GriffinPeter Griffin’s The Theory of Blackjack (First published 1979)Blackjack Hall of Fame

Why? This book is considered the bible on the mathematics of blackjack. Probably not required reading unless you are really interested in math.  None the less, this book included ground breaking information on the game including the most complete basic strategy ever published. Peter Griffin.

[ BUY ON AMAZON: The Theory of Blackjack ]


World's Greatest Blackjack Book by Lance HumbleLance Humble & Carl Cooper’s The World’s Greatest Blackjack Book (First published 1980)

Why? Introduces the very popular Hi-Opt I and Hi-Opt II card counting systems.  The Hi-Opt counts did not assign a value to the Ace, thus requiring that they be side-counted. Lance Humble and Carl Cooper.

[ BUY ON AMAZON: The World Greatest Blackjack Book ]

Million Dollar Blackjack by Ken UstonKen Uston’s Million Dollar Blackjack (First published 1981)Blackjack Hall of Fame

Why? At the time, this was one of the most complete books on advantage play blackjack ever published. Discussed everything from card counting to team play. Included the Uston Simple Plus/Minus, Uston Advanced Plus/Minus and Uston Advanced Point Count systems. Also, chapters on the art of single- and multiple-deck play, team methods, front-loading, spooking, cheating, getting barred and tournament blackjack. Ken Uston.

[ BUY ON AMAZON: Million Dollar Blackjack ]

Blackbelt in Blackjack by Arnold SnyderArnold Snyder’s Blackbelt in Blackjack (First published 1983)Blackjack Hall of Fame

Why? Introduced the easier unbalanced Red Seven count and the two-level Zen count. Topics included depth charging, money management, the true count, camouflage techniques, toking guidelines, hole card play, cheating, team play and the effect of table conditions. Written by the editor/publisher of the outstanding card counters resource, Blackjack Forum magazine. Arnold Snyder.

[ BUY ON AMAZON: Blackbelt in Blackjack : Playing 21 as a Martial Art ]

Blackjack For Blood by Bryce Carlson Bryce Carlson’s Blackjack For Blood (First published 1992)

Why? The best part of this book is Bryce Carlson’s insight into what I call the art of twenty-one. Introduced the advanced ace neutral two-level Omega II count.

[ BUY ON AMAZON: Blackjack for Blood ]

Knock-Out Blackjack by Olaf Vancura and Ken Fuch Olaf Vancura & Ken Fuchs’ Knock-Out Blackjack (First published 1996)

Why? Introduced the very popular K-O unbalanced card counting system. This count ranks as one of the top single-level counts available to players today. The K-O card counting system eliminates the mountain of mental arithmetic necessary to win at blackjack. Olaf Vancura and Ken Fuchs.

[ BUY ON AMAZON: Knock-Out Blackjack ]

Blackjack Attack by Don Schlesinger Don Schlesinger’s Blackjack Attack (First published 1997)Blackjack Hall of Fame

Why? The ultimate blackjack reference book for professional players. Covered topics such as back-counting the shoe game, betting techniques and win rates, evaluating new rules and bonuses, statistical insights, the “Illustrious 18”, the “Floating Advantage”, team play, camouflage, risk of ruin, and more. Don Schlesinger.

[ BUY ON AMAZON: Blackjack Attack: Playing the Pros’ Way ]

Burning the Tables in Las VegasIan Andersen’s Burning the Tables in Las Vegas (First published 1999)Blackjack Hall of Fame

Why? This was the long awaited sequel to one of the best-selling blackjack books ever written, Turning the Tables on Las Vegas. Included powerful camouflage strategies to avoid detection by casino staff while card counting. Ian Andersen.

[ BUY ON AMAZON: Burning the Tables in Las Vegas ]

Blackjack BluePrint by Rick BlaineRick Blaine’s Blackjack BluePrint (First published 2006)

Why? Think of this book as everything you need to know about team play that Ken Uston never told you. Anyone even considering joining a team must own this book. Revised and updated in 2014. Rick Blaine.

[ BUY ON AMAZON: Blackjack Blueprint: How to Play Like a Pro … Part-Time ]


Modern Blackjack by Norm WattenbergerNorm Wattenberger’s Modern Blackjack (First published 2009)

Why? This two-volume book is massive. It was written by the author of the outstanding Casino Verite suite of blackjack software which he used to provide highly detailed information and analyses of every aspect of the game. Norm Wattenberger.

[ BUY ON AMAZON: Modern Blackjack Second Edition Volume I ]
[ BUY ON AMAZON: Modern Blackjack Second Edition Volume II ]

Copyright © 2020 All Rights Reserved
Michael Dalton / Blackjack Review Network



1 Michael Dalton is author of the Encyclopedia of Blackjack and editor/publisher of the Blackjack Review Network. Looking for other good blackjack books? Check out the product review section of the Encyclopedia of Blackjack and the online book catalog on this site.

The Art of Twenty-One

by Michael Dalton

The Art of BlackjackThe key to success for most blackjack card counters is the act.  It is the very essence to the art of twenty-one.  It is what distinguishes the average card counter who has trouble finding a good game and the blackjack expert who manages to make the game good.

A player once told me that he gave up card counting because every time he raised his bet the dealer would shuffle up on him and intimidate him by trying to deal faster.  If this player had used some common sense and not acted so obviously like a card counter he probably would have been left alone.  A good card counter should be able to keep up with the fastest dealer, however, if you are having issues with this, here is a tip:  Just play your hand slower and think about every decision.  Don’t let the dealer dictate the speed of the game!  To be successful at twenty-one you not only have to be able to count cards you must develop a unique style of play that is different from what the casinos are looking for.


All books recommend that players not drink.  All casinos realize that drinking affects judgment and card counting and drinking don’t mix.  Therefore, players who drink are not a threat to the casino and more likely than not they are preferred to your average player.

I also recommend that you don’t drink when you play but no one ever said you couldn’t pretend to drink.  I will often order a Heineken or dark bottle beer only to empty 95% of its contents out and refill it with water.  This completes my full gambler/tourist act.  It doesn’t cost much since you can always refill your beer bottle with water.  Just don’t try carrying your beer bottle from casino to casino.  You might get arrested!


Many card counters prefer to play early in the morning (3-6AM) because they tend to get better games and there are less crowds.  However, most casinos also realize this.  If your schedule calls for early morning play, try not grooming yourself when you wake up.  In other words, don’t shave or shower.  From the casino’s point of view, you will appear to be a player that has stayed up all night and thus less of a threat.  Some players push this to an extreme!  I once ran into an fellow card counter1)Yes, a card counter can usually identify another card counter in the casino.  I remember the first time this happened to me many years ago.  I was at the Silver City Casino in Las Vegas playing a $1 minimum single deck game.  It only took about 15 minutes.  I didn’t like the fact that this player was practically mirroring my bets.  I later discovered after talking to him that he was using one of Lawrence Revere’s Point Count Systems.  Recently, I found a fellow counter playing much higher stakes on a cruise ship.  He was betting $50 to $500 or so.  I watched him for a few days and after being convinced he was accurately keeping the count, I would occasionally jump into his game betting my maximum bet whenever he had $500 or more on the table. who attempted to block seats while smoking, drinking and essentially acting like an idiot.  


One give-away that you may be a counter is that you sit quietly and stare at the cards.  You need to get over this!  A glance is all that should be necessary to accurately pick up the count. You should be counting cards as efficiently as possible – counting cards in groups of 2, 3 or more and canceling cards out.  For minus counts, never think “minus X”.  Always think “my X” or something simpler.  It is shorter and more efficient.  If you get distracted for some reason, have a method to “lock in” the count.  Using chips or a chip location has worked for me. 


Probably the best advice I can give to increase longevity in the casino is the ability to talk with other players, the dealer and pit personnel while accurately keeping the count and playing perfectly.  For more ideas on putting on the perfect “act” be sure to read The Blackjack Player’s Guide to Idiot Camouflage and Ian Andersen’s book below.

Burning the Tables in Las VegasIan Andersen’s Burning the Tables in Las Vegas (First published 1999)Blackjack Hall of Fame

Why? This was the long awaited sequel to one of the best-selling blackjack books ever written, Turning the Tables on Las Vegas. Included powerful camouflage strategies to avoid detection by casino staff while card counting.

[ BUY ON AMAZON: Burning the Tables in Las Vegas ]

Copyright © 1992 – 2020 All Rights Reserved
Originally published (in part) in the Summer 1992 issue of Blackjack Review Magazine

BLOG MENU 2)The Art of Twenty-One: Blackjack and Smoking Cigars Background image courtesy of Michael Godard, the “Rockstar of the Art World”.


1Yes, a card counter can usually identify another card counter in the casino.  I remember the first time this happened to me many years ago.  I was at the Silver City Casino in Las Vegas playing a $1 minimum single deck game.  It only took about 15 minutes.  I didn’t like the fact that this player was practically mirroring my bets.  I later discovered after talking to him that he was using one of Lawrence Revere’s Point Count Systems.  Recently, I found a fellow counter playing much higher stakes on a cruise ship.  He was betting $50 to $500 or so.  I watched him for a few days and after being convinced he was accurately keeping the count, I would occasionally jump into his game betting my maximum bet whenever he had $500 or more on the table.
2The Art of Twenty-One: Blackjack and Smoking Cigars Background image courtesy of Michael Godard, the “Rockstar of the Art World”.

Bill Zender Newsletter – Down Under Blackjack and Emails

Bill ZenderBill Zender. 1)EDITOR NOTE: Post below updated in 2023 to reflect Bill Zender’s email responses including Down Under Blackjack.

Answers to Important Emails

Correcting a loop-hole in their manual multiple deck shuffle

Below is the follow up to an email I received two months ago.  The casino executive contacting me had been watching a BJ customer punishing him with occasional large bets in the casino’s multiple deck shoe game.  The game was a higher limit table, and was hand shuffled.  Base on the original email I noted that the shuffle they used did not incorporate a deck “strip” in the shuffle procedure.  When questioning this oversight, I was told that any shuffle, or change to the shuffle, had to be approved by the gaming regulator.  I advised the casino executive that it was imperative that the casino insert a deck strip during each card grab selection where the cards are riffle shuffled.  I received the following reply the middle of July;

Reply by the casino executive:

I just wanted to drop you a quick follow up…

We obtained approval to change the shuffle procedure to incorporate a more standard riffle, riffle, strip riffle for all cards during the shuffle, and…

Our most recent suspected AP and associate returned, checked into a comped room, bought in for a large amount, observed the shuffle, played utilizing a minimal bet spread redeeming all the match plays available to him, retired to his room with his associate cashed out a short time later checked out of his room and left property in an Uber.  I can’t thank you enough for your insight and recommendations.

Additional comment on this subject:
Regulators need to understand that for the most part, they do not possess the knowledge needed to judge and dictate various procedures required to make their casino providers’ games safe from scams and advantage play.  I appreciate the job done by regulators, and I respect their watch-dog position for protecting the industry, however many regulators need to lose their egotistic attitudes, admit they don’t know the best method for conducting certain tasks and procedures, and seek outside expertise.

No commission Pai Gow Poker Variations

Email from a Director of Casino Games:
Have you ever seen a commission free PG that offered player banking?  I can’t think of anything that would vary the math away from the house and that would be different from a standard commission game.  In the game in question a Q High Pai Gow for the Banker results in a push of all Pai Gow hands.  We would just simply add a rule that states if a player is banking and the dealer’s hand is a Q high Pai Gow the “Banker’s” hand would then always push with the dealer.   Have you even seen a casino do this?  
Thanks for any input.  Also thanks for your input a few month ago about Baccarat.  Our game is up and running now and doing quite well.  Hope this finds you and yours well. 

My Reply:
There are versions of “Pai Gow Poker” where the game is dealt without charging the customer a commission on winning “player” bets.  I have attached a PDF of a slide I use in my math presentations.  It show three forms of PGP where commission is not charged on the customer’s winning player bets.  Remember, the banking customer is either prohibited from banking or still charged a 5%commission on winning banking results.  These games are based on a joker being used wild for straights/flushes, and as an Ace otherwise.

  1. You can change the standard PGP game to commission-less, however it reduces the H/A% from 2.7% down to 1.23%.  All customers banking results are still charged 5%, and are subject to the standard H/A% of 0.12%.
  2. In “Commission Free” PGP the bank pushes ALL bets with a 9 high (or less) low hand (based on house way).  Because a “9-high” low hand can be manipulated, customers are not allowed to take the bank.  House H/A% of 2.5%.
  3. In “EZ Pai Gow Poker” the bank pushes ALL bets with a Queen high (or less), and the customers can still bank, but are charged a 5% commission on all winning bank outcomes.  House H/A% of 2.47% on the player hands, and I’m not sure about how the customer bank is handled.  

That should help answer that question.  I do believe that the Commission Free PGP and EZ PGP games are both proprietary (if the patents are still good), and you will have to contact with the vendors.

Questions about “Down Under Blackjack”

There has been a lot of interest in the past several years for different versions of blackjack.  The most popular by far is “Spanish 21” which is a copy of Australian Pontoon.  The next popular variations is the BJ series of “push 22” designed and patented by Geoff Hall.  More lately, is the Dave Wisler creation of Down Under Blackjack. For those who are unfamiliar with the game concept, Wisler has developed a hole-card peeking system that actually informs the player about the approximate value of the dealer’s hole-card on every hand.  This information is broken into three different categories of “Blue” (low 2-5), “Red” (medium 6-9), and “Gold” (high T & A).  The information is helpful to a point, but it still presents the player with a number of hole-card value possibilities that the player has to consider before reaching a hand play decision.  Following is my Q&A with this executive.

Executives Email:
I am sure you know Dave Wisler from Masque Publishing.  He just got “Down Under Blackjack” approved in this State.  I remember looking at the game last fall at G2E.  Please share your opinion when you have time. 

My Reply:
In reality, it’s another BJ variation in the same fashion as Geoff Hall’s “22 push” series of BJ games.  The H/A% will be the same as your standard BJ games give or take a small percentage, but it will cater mostly to lower limit players (and the occasional high limit player who is temporarily curious).  You should also get a high degree of side betting action on the Match the Dealer which should raise the return high enough to easily offset the proprietary fee.  To my knowledge, DUBJ does not pose any new game protection issues, however the game does require the floor/surveillance to understand three different basic strategy charts correlating to the three hole-card categories of small hole-card, medium hole-card, and high hole-card.  That is my two cents on the game.  I hope to see you at one of the conferences this fall.

Note: DUBJ strategy charts can be found at: https://wizardofodds.com/games/down-under-blackjack/
Knowledge SharingQuestions???
I’m always available to answer your questions. If you have any questions on gaming; don’t hesitate to contact me through email.  I answer close to a half dozen emails on gaming every day wzender@aol.com. 
Cheers and good luck.
Bill Zender and Associates


1EDITOR NOTE: Post below updated in 2023 to reflect Bill Zender’s email responses including Down Under Blackjack.

How Casinos can be Sure a Person is Counting?

By Bill Zender

October 2, 2017

Dear Friends and Business Associates,

Galaxy GamingI will be attending this year’s G2E tomorrow, October 3rd, and Wednesday October 4th.  Since I am on the Board of Directors, I will be “hanging out” primarily at the Galaxy Gaming booth 4208.  The convention area is like a small city so be sure to grab a map and look for the booth’s location.  Come by and see Galaxy’s sales “guru” Dean Barnett, or the company’s new CEO, Todd Cravens.  A longtime associate of mine, Gary Saul, will also be working the booth.  While you are there, ask Gary to show you a card trick or two.

Part of the time I plan to visit the Techart booth 3535.  Techart is the maker of the MAXTime hole card reader.  I will be there visiting two of my good friends, Bob Del Rossi and Artie Miller.  Artie is one of the winners of the World Game Protection Conference’s “Lifetime Achievement” awards for his contribution to the gaming industry and game protection.  Not a bad place to spend time.
If you catch me around 2PM on Wednesday, maybe you can join me while I walk the exhibition floor.  I like to tour some of the booths and look at new and improved products that involve table game productivity and protection.  It’s usually a fun table game “walk-about”, and you never know what you might find.  Hope to see you there.

How can a casino executive be sure a person is actually counting?How can a casino executive be sure a person is actually counting?
This is the third portion of the Card Counting Protection series I have detailed in my monthly eBlasts.
Before backing anyone off a game, changing deck penetration, or limiting the player’s bet spread, one needs some reasonable tools to determine that the suspected person is a professional level card counter.  The executive or surveillance professional who is conducting the evaluation needs to use some type of analytical tool that will accurately determine the possibility that the suspected player is in fact a card counter, and also provide the evaluator with a printout, or hard copy of these results.  This hard copy can be used to back up the evaluation, and to later serve as proof that the suspect was in fact counting cards.  How many times have your department taken action against a player for counting cards, only to have the legitimacy of the “back off” questioned by a host or casino marketing afterwards?  In addition, this evaluation tool can serve as evidence as to why a player wasn’t backed off even though the customer was a big winner.

There are two tools that I recommend casino executives can utilize.  The first tool is a blackjack software package that analyzes a blackjack player’s skill and betting level.  There are two very effective software packages available to the casino executive.

Information regarding the person’s actual play is entered into the program either manually or through a voice recognition program.  After enough hands are entered into the program, it is queried by the analyst. The program will indicate to what degree the information of the betting and hand decision patterns indicate the suspected player is counting cards.  It’s important when using any type of card counter verification method that several decks or shoes are analyzed in order to calculate the best possible decision outcome.  The primary drawbacks to the software are in its inability to take into consideration other influences in the game, and the hefty price tag that comes with the software.

The second card counting analytical tool is a much lower tech, less expensive system that accomplishes the same goal just as accurately.  This is known as “charting”.  A number of surveillance departments already use some form of charting a customer’s plays as a means to better understand the person’s play characteristics.  Charting is most effective as a card counter catch tool when the evaluator uses it to determine the breadth of the suspect’s betting spread, and the suspect’s betting correlation with the true count of the cards.  In order to win money counting cards, the counter must wager significantly more money when he has the advantage, and as little as possible when the casino has the advantage.  If the charted observations of a suspected player indicate that he is utilizing a large enough bet spread, and increasing his wagers to the larger bet level when he has a mathematical advantage of 90% or greater, then the evaluator has enough information to safely rule that the suspected player is counting cards.  As noted when using a software package to evaluate a suspected card counting play, it is prudent to watch no less than 4 decks or shoes before making the final decision about the player.

Following are points that need to be followed to ensure the casino executive or surveillance professional is making the correct analysis regarding a suspected blackjack player’s long-term ability to be a card counting threat to the casino’s bankroll.

  • Don’t automatically assume that a winning blackjack player is counting cards.  There’s a greater chance he’s a desirable player who is running luck.  Assuming a winning person is a card counter will also cloud one’s ability to look for other more costly problems such as advantage play techniques and cheating.
  • It’s a must for a successful card counter to have an effective bet spread, especially in the six deck and eight deck games (don’t forget the 6:5 single decks).  Without the necessary bet spread range, the player cannot gain an advantage counting cards.
  • Never cut the shoe in half on a player to look for a reaction.  You need to analyze the play before taking any action. If you cut the shoe in half on a desirable blackjack customer, he could get upset and never play at your property again.
  • Don’t rely on a decision that is made by one of your floor executives or surveillance operator who is not at least on a semi-professional card counting level.  I’ve known a number of executives who claim to know card counting inside and out, but when asked, can’t answer simple questions such as the mechanics of true count conversion and hand strategy deviations.  Take the Ouija board out of the equation.
  • Whether your organization uses a counter catcher software package or elects to chart the play, always look at several shoes or decks before making a decision about a player.  I prefer looking at several examples where the count is mostly minus throughout the shoe, as well as several examples where the count is mostly plus.  Taking your time to compare the extremes will help eliminate the chance of getting “false positives”.
  • Do not take action against a suspected player until you are 100% sure the player is counting cards.  If you wish to error on the side of the casino, take your time and make sure you correctly identify a customer as a long-term card counting threat before disrupting his play.

For more information on charting, please feel free in contacting me wzender@aol.com.
Seminars and workshopSeminars and Workshop

CGA Conference – Protecting Cardroom Table Game Integrity – Monday, October 16th, 2017 – Hard Rock Lake Tahoe Nevada

I just finalized the material in my presentation at this year’s California Gaming Associations meeting at Lake Tahoe which will be held October 14th through 16th.  I will be presenting a 60 minute session on Protecting Cardroom Table Game Integrity.  The presentation will include different methods that are being used to cheat table games in the California card rooms.  This not only concerns poker games, but also the rotating bank games such as baccarat, blackjack, and pai gow poker.  The following topics are:

Capturing playing card sequencing
It has become a common attack for cheaters to attempt to capture shuffled card sequencing on high limit games.  The captured sequences are used to learn winning hands occurrence before making wagers on the game.  The primary game of attack is baccarat, but this techniques have also been used in higher limit blackjack.

Switching hands in Pai Gow Poker
The game of Pai Gow Poker is played with seven cards placed into two separate combinations.  What if a player was able to place a large wager on one hand, but have the ability to choose from two separate seven-card hands?  In this segment the participants learn how a husband and wife team were able to switch hands without the dealer, floor supervisor, or the designated player questioning their actions.

Protecting Bad Beat and High Hand Jackpots in Poker
Ever have an unusual cluster of “Bad Beat” jackpots occur over a short period of time?  Was it a statistical anomaly or was it created by cheaters working with the dealer.  Learn how a group of cheaters would set up a “Bad Beat” cooler deck, and what your surveillance personnel need to look for specifically.

Player collusion in Pai Gow tiles
What would happen if a group of Pai Gow players shared their tile information with each other?  Do you know what to look for?  Learn the primary indicators of a player collusion scam in Pai Gow tiles, and learn what you can do to prevent them from taking advantage of your Pai Gow customers.

At this time I’m not sure whether the presentation will be held in the morning or afternoon session.  I will inform everyone of that schedule in next month’s eBlast.  I believe you have to be a member of the CGA in order to attend.  For more information please contact California Gaming Association, Joe Patterson, Executive Director, (916) 297-4822, www.californiagamingassociation.org/.
Cutting Edge Table Game Seminar – Tuesday, November 14th, 2017 – Paris Resort – Las Vegas Nevada
I will be presenting once again on the first day of the Cutting Edge Table Games Conference.  This is another conference in which I like to present.  This year’s session covers a lot of the information presented last year, however I have included the much talked about topic of “Rolling Chip/Dead Chip Programs”.  I always like to update my information and slip in topics that I feel need to be covered regarding the operation of table games.   Following is an outline of this year’s session:

I.     Table Games Mathematics – 9:00 AM to 12:00 PM

  • House advantage of major casino games
  • Mechanics that drive hold percentages up (and down).
  • Table game procedures that waste time and money
  • Understanding how to optimize side bets
  • Managing table game minimums to yield the best returns
  • Calculating risk range for table games and table limits
  • Determine the best metrics to use in player tracking systems.
  1. Understanding Rolling Chip/Dead Chip Programs – 12:00 PM to 12:30 PM & 2:00 PM to 3:00 PM
  • What is a Rolling Chip Program?
  • Understanding Multi-play Chip mathematics
  • Percent of the Rolling Chip Buy-in that you can use for Player Reinvestment
  • Points needed before initiating a Rolling Chip Program

III.     The Cost of Table Game Promotions – 3:00 PM to 5:30 PM

  • Effect of changing table game rules
  • Cost of “value added” promotions
  • Cost and effect of using Match Play
  • Cost of using Single Play Promo Chips
  • The pitfalls of using play-till-you-lose Promo Chips
  • Problems with discounting player losses
  • Using an Adjusted T-win model

The conference will be held at the Paris Resort in Las Vegas, November 14th through 16th.  For more information please go to their website www.tablegamesconf.com/ .  I hope to see you in Las Vegas in November.

On the moveOn the Move
It appears I will have at least one week open each month to the end of the year.  I always have time open to conduct winning player reviews.  If you have any questions, let me know wzender@aol.com
California Gaming Association Presentation
Northern California
Albuquerque NM
Table Game Cutting Edge Las Vegas
BOD Meeting Arizona
Central California
Please let me know if I can provide one of my services for your organization.
I’m always available to answer your questions. If you have any questions on gaming; don’t hesitate in contacting me through email.  I answer close to a half dozen email on gaming every day wzender@aol.com
Cheers and good luck.
Bill Zender
Bill Zender and Associates

What are the top blackjack play variations…

… that yield the greatest gain for card counters?

By Michael Dalton

Illustrious 18 play variationsThe top play variations for blackjack card counters are knowing when to take insurance and stand on 16 versus a dealer’s 10. The top 18 plays, known as the Illustrious 18 are show in the table below in order of importance 1)The gains cited are associated with a player spreading bets from 1 to two hands of 6. These are the average gains for flat betting multiplied by the average bet one has out when he makes the departure.

These top plays were recommended by Donald Schlesinger in an important study published in the September 1986 issue of Blackjack Forum. The study suggested that players might just as well forget all of the other play variations and stick to the top 18 plays since this is where most of your profitable gain is going to come from. In December 1995, Schlesinger continued his study to include the top surrender plays which have become known as the Fab 4:

Fab 4 play variationsThe analysis and discussion that presented the above conclusions can also be found in Donald Schlesinger’s book Blackjack Attack, which was first published in 1997. This book includes some of the most important technical insights into the game of blackjack ever published.

Play variations (variations you make from basic strategy) are what card counting index numbers are all about. They can be very important in single deck and become less important as decks are added to the game. Each card counting system assigns unique index numbers for these plays. For example, in the Hi-Lo system the Insurance index for a 6-deck game is 3.0. This means that the player’s best technical move is to take insurance if the “true” count is 3 or higher. If the player has a 16 versus a dealer 10 the Hi-Lo index number is 0. This means that the player should “stand” if the count is greater than or equal to 0 and “hit” otherwise.

Although these top play variations are technically correct you may have difficulty implementing some of the plays from a camouflage point-of-view. For example, the casino pit will often watch players who correctly take insurance on bad hands and decline insurance on good hands. Splitting ten valued cards is another give-away that you may be counting cards. The saying is that only two types of players split tens — idiots and card counters. If you don’t look like an idiot at the table they may just suspect that you are counting cards. Many counters have given up splitting tens for this very reason. However, insurance is too valuable a play variation to ignore. If you need to mix-up your play I would suggest you consider taking even money on small bets occasionally when the count does not justify it.

When you are ready to add more play variations to your repertoire you may want to consult Peter Griffin’s book, The Theory of Blackjack and the article “The Most Important Plays in Blackjack” published in the Summer 1992 issue of Blackjack Review Magazine.


[ The Illustrious18 ] [ The Fab 4 ] [ The Hi-Lo Count ] [ Blackjack Apprenticeship: What are Blackjack Deviations ]

Copyright © 1994 – 2020 All Rights Reserved
FAQ 12: Originally published in Volume 7 Issue 1 of Blackjack Review Magazine



1The gains cited are associated with a player spreading bets from 1 to two hands of 6. These are the average gains for flat betting multiplied by the average bet one has out when he makes the departure.

What is the best card counting system?

By Michael Dalton

Dealer Dealing BlackjackThere is no such thing as a “best” card counting system unless you are a robot, in which case you would be keeping track of each and every card. Humans, on the other hand, have limitations. For shoe dealt games, all card counting systems perform within a tenth of a percent (or so) of each other. For single deck games, a balanced multi-level count with an ace side-count can show a significant theoretical improvement over unbalanced and single-level counts, however, the player runs the risk of mental fatigue and errors. The bottom line for most players is that “simple is best!” I recommend the Hi-Lo, Red Seven, K-O or Zen for shoe games and the Hi-Opt I, Zen or Omega II for single-deck. What is the best all around count? I like the Hi-Lo because it allows me to concentrate on more important things… like convincing the pit boss that I am a loser!

Obviously, the answer to this question is not as easy as it appears. Several approaches have been used in the past to evaluate card counting systems. One analytical approach is the calculation of several performance parameters (e.g., playing, betting, and insurance efficiencies). The results are then used to approximate the potential of one system over another. Another approach that is used is to simulate each system against typical game conditions on a high speed computer. Simulations can provide an accurate real-world estimate of the advantages and win-rates that are possible in playing a particular system.

However, the problem with coming up with a ‘best’ card counting system is that you can always come up with a ‘better’ card counting system. Instead of a single-level 1) A single-level count assigns point values in such a manner that the non-zero point values are the same in absolute value, namely +1 or -1. The single-level Hi-Lo count, for example, assigns 2 – 6 as +1, 7 – 9 as 0, and Tens and Aces as -1.  ‘unbalanced’ count you could assign more accurate point values to each card and determine true counts by the exact number of decks or cards remaining. You could improve ‘playing’ efficiency by assigning a ‘zero’ to the Ace and side counting each of them. You could also side count other cards such as 7s, 8s, and 9s thus improving your play against specific hands. You could also incorporate play variations (changes to basic strategy) based on specific counts by remembering ‘every’ index number for ‘every’ play possible. To improve the accuracy of your insurance decisions you could also keep a separate count of all the tens in the deck or shoe. Of course, you don’t want to forget all the ‘practical’ advice each system offers in regard to betting, playing, camouflage, and other tips and tricks of the trade.

Peter Griffin, author of the classic text The Theory of Blackjack, wrote “If one’s ambition is to raise overall strategic efficiency beyond the 70% level, perhaps as high as 90%, it is imperative that the primary system be quite simple and hence allow great flexibility for incorporating several auxiliary, independent sources of information.

I believe the above comment was one of the most important suggestions ever made about card counting. Griffin suggested that it may be better to keep your base count simple to allow your brain the ability to perform other tasks and to utilize other sources of information. These other sources of information can often improve the potential of a single-level count over an advanced 2- or 3-level count that doesn’t use this information. This information includes side counts, shuffle tracking, ace location strategies, key card techniques, and dealer errors. My own experience at card counting has shown that Griffin was probably right.

Human error is another reason to keep it simple. The most advanced card counting system in the world is not going to do you any good if you can’t play it accurately. Thus, the best card counting system may be one that perfectly balances theoretical power and your human ability to execute it accurately.


Copyright © 1994 – 2020 All Rights Reserved
FAQ 11: Originally published in Volume 6 Issue 4 of Blackjack Review Magazine



1 A single-level count assigns point values in such a manner that the non-zero point values are the same in absolute value, namely +1 or -1. The single-level Hi-Lo count, for example, assigns 2 – 6 as +1, 7 – 9 as 0, and Tens and Aces as -1. 

Isn’t card counting illegal?

By Michael Dalton

JusticeNo! It’s no more illegal than using your brain to add up how much money you have in your wallet. But that doesn’t mean they won’t kick you out for trying! In most casinos in this country casino management has the right to bar players for any reason they want. Most casinos are considered private clubs and if they don’t like you they can ask you to leave. In Atlantic City, New Jersey, casinos cannot bar you but they are allowed to implement specific countermeasures that will reduce if not eliminate any profit potential you previously might have had.

If casino management suspects a player is a threat to its bankroll they will usually implement several countermeasures, which include reduced deck or shoe penetration and shuffling up. Psychological tactics are also often attempted to distract the player and to convince him or her to leave. Some less than reputable casinos may even implement cheating countermeasures — such as preferential shuffling. If all else fails, a suspected card counter may be asked to leave and/or to refrain from playing blackjack any further.

In Nevada, the courts have made it clear that card counting is legal. During a cheating case in 1983, which involved a player who was crimping cards to gain an advantage (definitely cheating), the court made an interesting statement: “By way of contrast, a card counter — one who uses a point system to keep track of the cards that have been played — does not alter any of the basic features of the game. He merely uses his mental skills to take advantage of the same information that is available to all players.” As I. Nelson Rose, author of the book Gambling and the Law*, states — “The card counter is playing by the rules of the games, as set up by the casino regulators and the casinos themselves.”

When I play blackjack I sit there and watch the cards. I use my brain to make decisions. I risk hard earned money on individual outcomes that are far from certain. If card counting were considered cheating then casinos would probably have little signs stating, “Thinking in this casino is forbidden! Players found using their brain to make decisions will be asked to leave.”

In a similar vein… if a dealer unintentionally flashes his hole card and a player sees the card, would it be considered cheating for the player to make use of this information? Of course not! The player is still playing by the rules of the house and is simply using all the information that is presented to him. Of course, if the dealer is intentionally flashing his hole card then that would definitely be considered cheating in every court in this country and both player and dealer could be charged with an illegal act. [If the dealer were intentionally flashing it to a third part confederate, an innocent bystander who didn’t know he was flashing intentionally wouldn’t be guilty of anything; but, he might have a hard time proving he wasn’t in on the caper!]

Although card counting is not illegal from a technical point of view, in some countries you still might find yourself behind bars, forced to return monies won, or worse! You are on your own in some of these third world countries where the law is often undefined.

Professional players are at the greatest risk because of the betting level they play at and because of reporting agencies such as Griffin. The Griffin Detective Agency of Las Vegas, Nevada publishes a monthly list of blacklisted players to casinos around the world. This list includes suspected cheats, con artists, card counters, and even card counting team members. If you are in a foreign country and they notice that you are on this list they may just throw you in jail… not bothering to make the distinction between being labeled a cheat and someone who uses legal card counting techniques.

*Editor note:  Since this article was originally written, I. Nelson Rose and Robert A. Loeb had collaborated on the excellent book Blackjack and the Law.  Also, Robert Nersesian’s book The Law for Gamblers is a must read.  Another good article on whether card counting is legal or not is on Norm Wattenberger’s site IsCardCountingIllegal.com.

Colin and Ben from Blackjack Apprenticeship discuss “Is Card Counting Illegal?

Copyright © 1994 – 2020 All Rights Reserved

FAQ 10: Originally published in Volume 6 Issue 3 of Blackjack Review Magazine


How much money can I expect to win if I count cards?

By Michael Dalton

Money in the HandTo be perfectly honest, you stand a fair chance of losing money in this game! It all depends on your skill level, bankroll, the level of risk you are willing to take, and the quality of games you play. Assuming you balance all of these factors, the theoretical long-run advantage a skilled card counter can obtain is between 0.5% and 1.5%. This is not to say a player can not have a greater advantage on any individual bet or circumstance, however, in the long run a player can expect to win about 1% of the total sum of his “action”. Therefore, if your average bet is about $25 and you play 75 hands an hour you might expect to make about $19 an hour. But be forewarned… your mileage will vary!

I generally tell players that they can expect a winning streak as often as a losing streak in this game. You will often go for days, if not weeks and possibly even months questioning your ability playing blackjack because of losses. But you will also go through periods where you will feel invincible and unable to lose. During both of these periods your discipline will be severely tested. In the long run, if you are a good player, are not being cheated, and can get away with it, you should show a profit.

Basic card counting is not the only legal way to make money at this game. You can milk the comp system. You can use casino coupons. You can shuffle track. You can join a team. You can take advantage of dealer errors and mistakes. You can exploit casino promotions and new variations. I would estimate that it is possible to achieve at least a 2% win-rate in blackjack if you are an advanced player and use some of the above tricks and methods.

Milking the comp system is a true art form. It is getting tougher these days due to computers, however, it is still possible to achieve more than your fair share of comps. According to Max Rubin, author of Comp City, more than half a million dollars worth of complimentaries (free drinks, food, rooms, shows, limo rides, airfare, golf, and more) are handed out every day in Las Vegas. On weekends and holidays, comps climb into the millions.

Casino coupons are also wonderful! From those simple lucky buck coupons and free meals to discounts off your room — if you use them often I guarantee they will become like extra cash in your wallet. You can often find good coupons in local newspapers and magazines and through publications such as the monthly Las Vegas Advisor.

Shuffle tracking 1) Shuffle tracking is an advanced form of card counting.   A complete tutorial and expose of shuffle tracking can be found in Arnold Snyder’s 3-issue Shuffle Tracking Series from Blackjack Forum Magazine.   is an advanced area of card counting and if mastered you can expect to easily double or triple your expectation. That is assuming you find “trackable” games of course. Unfortunately, it is becoming harder and harder to find these games.

Some players join teams to allow them to reduce monetary fluctuation and play with a bigger bankroll. Also, non-counters can be used effectively on a team. Theoretically, you can make more money by playing on a team, assuming you trust everyone to accurately report wins and losses.

Dealer errors and mistakes happen occasionally and I would estimate they add a small percentage to my overall win-rate. It may not be much but I will take every edge I can get.

Your greatest expectation can come from exploiting casino promotions and new games and variations. Casinos have offered 2 to 1 on blackjack, early surrender, single deck with an edge, bonuses, jokers in the deck, etc. If you are sharp these games can be a short-time gold-mine.

But most important to how much money you can make in this game is the element of time. If you don’t play they don’t pay! Put in the time and everything else will take care of itself.

Copyright © 1994 – 2020 All Rights Reserved

FAQ 9: Originally published in Volume 6 Issue 2 of Blackjack Review Magazine



1 Shuffle tracking is an advanced form of card counting.   A complete tutorial and expose of shuffle tracking can be found in Arnold Snyder’s 3-issue Shuffle Tracking Series from Blackjack Forum Magazine.