Chapter 3: The Blackjack Formula

Blackjack Therapy by Clarke Cant

Chapter 3, the Blackjack Formula, revised and revisited.

My penname, Clarke Cant, began in 1980, when I went to the GBC bookstore, in Las Vegas, and I heard someone in back tell Howard Schwartz, “you know sometimes the math of blackjack seems to be such a waste. There is so much effort put into analysis of what is still a game.” I didn’t know who said it, but it sounded like GBC’s founder leading someone to their car out back.

Tom, who used to hold down the title of blackjack expert, before they hired Paul Keen, told me, “that was Stanford Wong you just missed going out the back.”

I wrote a letter later to Wong concerning a free newsletter offer that he had, added this phrase (paraphrased here):

“You ought to know that the math of blackjack is an example of sampling and partition theory that has applications in particle physics, where the mass of virtual particles changes their interaction with others, much the same way blackjack changes when dealt from a single deck, and then dealt with the same rules from a shoe…”

I didn’t know then but this technique was actually new in physics and would later be used at the CERN supercollider to estimate the mass of vector bosons.

When I got the envelope from Wong, my signature of my real name was garbled, when read by Wong, into Clarke Colbey(?) in the address. I realized just how bad my handwriting was. I was also teased by a friend about my libertarian views, and how my handwriting often varried the same way Lenin’s or Kant’s did. Hmm, Clarke Cant—and the name stuck.

One of the sample newsletters was only a month old and mentioned a new book by Arnold Snyder called, The Blackjack Formula and gave some examples from it. I wrote Arnold a letter pointing out that one example was wrong in that betting gains will not rise past a certain point, where otherwise you would have an impossible situation where betting gains were more for staying and playing through negative counts, than you would have table hopping.

Snyder wrote back that I was the second to point this out, Griffin (Peter, author of the Theory of Blackjack) was the first, and that he was the first only because his envelope was on top in the Snyder mailbox at home. Also enclosed was a comp copy (then selling for $100) of the Blackjack Formula. An exposition of that method of estimating your possible winning using a counting system is given in this chapter.

Years later he wrote several pieces in his magazine, Blackjack Forum, about the development of that original book. Bits of explanation were also scattered in several letters to me about difficulties in doing a California startup, when I wrote him about the difficulties in doing a Texas publishing startup, and had sent a prior version of this book to him, originally intended to be a companion to his Blackjack Formula. Later I received a permission letter from him to include the formula in my Clarke Cant Blackjack Utilities, published in 1985 as shareware for the Atari 8-bit (6502 based) home computers, and copyrighted with the Library of Congress. He also gave permission to use his Algebraic Approximation paper on deriving indexes directly from the effects of removal tables from Theory of Blackjack. His permission was included in the paperwork I sent the LoC and can be obtained, along with his letters about sharing the original manuscript, by anyone willing to send the LoC the copying fees (one controversy down but more fun to come). I only asked that my real name not be disclosed from anyone requesting my paperwork.

When the Blackjack Formula was first written there was little computer power available for all the simulations possible today. But virtually any game could be estimated for its return with most common calculators. The formula was also a bit more accurate than even Arnold gave it credit and with a little tweaking can be applied to today’s games with almost as much accuaracy as a 50 million hand simulation, and can still give results that are more than accurate enough to set your bankroll requirements to an accuracy that would not be noticably different for any player’s experiences.

But mostly I wanted to see this classic revised and updated because, as powerful as today’s simulators are, there are times in entering the casinos, and staying in their rooms, that we don’t want to have our computers with us ( and subject to being seized with all of our private details on their hard drives). The best example I can think of is the experience of a friend in the US Forestry Service, whose wife was doing contract programing for IGT, and had their laptop with them.

In a southshore Lake Tahoe casino, a malfunction occurred in one video poker bank of machines while he and his wife had their laptop out, waiting for their table in the coffee shop. His wife had been talking about working for IGT. He almost lost his daughter over the incident in that when the charges of violating Nevada’s device law were dropped, his hard drive had still been “gone over,” and nude photos of his daughter were found. His family had been nudists for generations and some family vacation photos were on the hard drive.

Each component of the blackjack formula will be discussed; some of those bigger pieces will be mentioned before some of the ingredients in that piece are explained to avoid cluttering details. This sort of top down explanation will be easier to understand. That means however that some variables will be given before they are labeled or explained. Be patient therefore as you read through this chapter and some of the others. Most labels will be explained, as will changes from the original formula. …all will come out in the wash…

  • BJF=Blackjack Formula result; BJF=BA+PA+RA+SA
  • BA=betting advantage
  • PA=playing advantage
  • RA=rule advantage
  • SA=starting advantage
  • H=spread, in this use it is the ratio of your average high bet to your waiting bet and is based on playing all hands and not table hopping.
  • P=number of players to your left; the original formula used the total number of players. This change is made to note how “depth charging effects,” from Snyder’s Blackbelt in Blackjack, compensate for playing with others at your table.
  • N=number of decks
  • LH=the upper limit of where your spread can be entered directly. If the ratio of your average high bet, to your waiting bet, is higher, use LH for calculating the BJF and your actual spread for things like your bankroll estimates.
  • BE=betting efficiency which is virtually the same as the correlation between the count you are using and the effects of removal for the type of game you are playing. There will be details on how to estimate this later.
  • SQR=square root
  • LH=38*[C-(P/30*N)]^2/SQR(N); the original formula used 20*N. Spreads that are more than this exagerate your edge. Use LH also if you are table hopping out when you don’t have an edge in the games you play.

Your BA is then:


The PA estimate of your playing expectation, for using playing indexes rather than basic strategy, is something that Snyder termed an example of his “jazz mode” of doing math. The formula is:


C is the fraction of cards remaining undealt when the dealer shuffles. The formula overall is more accurate with this estimate, rather than using an estimate for C based on the number of cards when the dealer begins his last round, which you might think of when you consider the simple fact that betting gains only come from the cards seen before the last round. Many elements of the BJF are not precise in themselves but where errors thankfully tend to cancel themselves out. This is one example of this.

  • CC=correlation coefficienct
  • IP= inner product

The inner product is the sum of each individual count value (or tag being the more up to date term) multiplied by the effect of removal, going card by card value. For the betting CC these values, tweaked at bit by Arnold Snyder, give good results for most games:

Ace       2       3       4       5       6       7       8       9       X (used here for all tens)

-.61     .38   .44   .55   .69   .46   .28     0   -.18   -.51

Add in the tens effect 4 times; there are 4 tens per suit.

For Hi Opt I these are the inner products:

Card     effect of removal Hi Opt I card value             inner product term

Ace         -.61                                 0                               0

2             .38                                 0                               0

3             .44                                 1                               .44

4             .55                                 1                               .55

5             .69                                 1                               .69

6             .46                                 1                               .46

7             .28                                 0                               0

8             0                                   0                               0

9           -.18                                 0                               0

X           -.51                                 -1                               .51 *4=2.04

The inner product is: IP=4.18

The CC formula is: CC=IP/SQR(SSE*SSP)

  • SSE=sum of the squares of the effects of removal, in this case SSE=2.818
  • SSP=sum of the squares of the values of your point count; for Hi Opt I SSP=8

For Hi Opt I the CCis just over .88; your BE is thus rounded to .88

The CC is not used directly for estimating the PE, but is used in a formula that Arnold Snyder has stated several times came from his jazz mode of doing math, with a model that made Peter Griffin cringe, but which gives very good estimates never the less. Snyder developed predictive effects of removal that derive from an count Griffin developed to give optimal single parameter results with values ranging up to 180. Here we have another instance of a math model that gives good results but in no way is precise in its ingredients, and which also gives results which are accurate enough that no one will ever be able to feel any impact of their errors.

These special effects of removal are:

Ace       2       3       4       5       6       7       8       9       X

.25       .3     .43   .62   .85   .61   .58     .22   -.26   -.90

SSE= 5.508

IP=6.11     CC=.920447984, OK that is a little more digits than the formula is accurate to, but what the hell.

The final PE is: PE=[1.405-(1-CC)]*CC/2

Everybody raise their hands who got .61 (rounded) for the PE for Hi Opt I.

The playing advantage now has all of its ingredients. I hope you can see how much easier it was to explain working from the top down rather than getting lost in some of the details.

The playing advantage requires some adjustment for some rules (just wait til we go into explaining a variable V we will use too). Bustout is an option that the Riverside in Laughlin used to have on some 6 deck shoes, where you would bet on a hand busting in one hit by using your counts infinite deck (later, be patient) index for insurance This rule adds a whopping 8/7 to your PA. If your game doesn’t include insurance (boo on the Cal-Neva in Reno Nevada for not having it) you subtract 1/7 of the PA. You can add 1/7 of the PA if your games has late surrender (well the Cal-Neva did have it for awhile). This late surrender gain presumes that you are at least using the “Fab 4” indexes that Don Schlesinger popularized (I usually use more).

Another adjustment is Snyder’s RA, or rule advantage, for more general reductions (usually) in your gains from rule changes. Your playing advantage is reduced for things like being limited in your double downs in d10 games (be patient) or when the dealer hits soft 17s.


Your SA is your starting advantage, or lack thereoff, as you begin playing a fresh shuffle (which is not the same thing as aproaching a continous shuffle machine unless it is just after new cards are put in, provided of course that the machine is allowed to cycle through a few times and it helps if the cards are “washed” by being spread about randomly before being loaded).

SA=DA+V; V is the variation from what used to be called standard strip rules but is now labeled DOA s17, which is double downs allowed on any first two cards, except the first two cards after the first hit when you split cards, and that the dealer stands on all 17s.


The Blackjack Formula originally gave V tables for 1, 2, 4, 6 deck games, but the advent of shuffle tracking has complicated things. Many times shuffle trackers in their play are playing segments that are odd deck sizes that wouldn’t be easy to fit into such tables. This either requires nearly 100 columns (which would resemble a certain…well skip it) or the solution here of 2 major columns and interpolation between them. Both the one deck and infinite deck Vs start at zero and change with each rule changes as given below:

             One deck V         Infinite deck V explanation (excuse?)

No d11       -.81                 -.73             cannot double on 11

No d10       -.52                 -.45             cannot double on 10

No d9         -.132               -.076           cannot double on 9

No sd         -.131               -.083         cannot soft double– want Pappy’s excuse?

No $A         -.16                 -.18             cannot split aces (not asses)

No nonA$   -.21               -.25               cannot split the others either

No re$       -.018             -.039             cannot split again

Re$A           +.03               +.08             do it again to that ace I drew

DA$             .14                 .14     this is the flavor if your game is otherwise DOA, double after splitting.

DA$ 11 only +.07               +.07            occasionally in Puerto Rico (where NY democrats go to find spare voters who get Musicals written about them)

DA$ 10 only +.05               +.05             but seriously if 10s but not 11s (obviously you add these if your flavor is in a d10 place where you can only double if you have at least a 10)

DD3+         .+.24                 +.22           can double after taking a hit

2to1 BJ     +2.32               +2.25           If this is only with one suit of ace or ten divide by 4; if this is in a matching suit divide by 4; if this is with a specific ace and ten divide by 16 (or go ask Stanford Wong and bug him—just kidding)

h17           -.19                   -.22             dealer stands on most 17s but hits soft 17s

These rules don’t fit the above chart:

Later surrender, abreviated LS usually, is +.022 in one deck, and +.069 in all multideck s17 games. LS in h17 games is +.036 in one deck and +.088 in all multideck games.

Early surrender—just a reminder, this is before the dealer is able to check whether they have a potential blackjack, is +.62 in all s17 games, and +.71 in all h17 games.

The rest of these V modifications apply to all 17 flavors and all numbers of decks:

Redouble, adds +.4 to your Vs. In this rule you can double your bet again (4X your original bet) if your double down card still gives you a double down recommended hand. This is just the rule if your double on a 9 draws a 2.

21 pushes a dealer blackjack adds +.17 if it allows ties with a dealer’s tenup blackjack and another .17 if you can tie an aceup blackjack as well (which only happens in games which take no dealer hole card until the players are finished with their options).

ReDA$ adds +.6; a test if you are understanding the abreaviations. It is a rare rule indeed.

ReDA$ just aces; which is the above when you can soft double and redouble if you split aces.

Joker declare in blackjack is where the pack has one or more jokers that you declare the value of when they are drawn as one of your first two cards. It adds: +4.2/joker/N

22 counts as 21 with coupon now I really mean ask Stanford Wong. His explanation is not too clear in Basic Blackjack. The sources of these tables are Basic Blackjack, Theory of Blackjack and some of my own simulations (and snide remarks). Actually this is from a one time coupon in the north shore of Lake Tahoe so there is little to worry about. The rule actually adds 14.6% of your bet with each coupon use.

To use this information to determine the V for your game add the infinite deck values and then the values that apply to one deck.. V1 would be your one deck figure, while Vi would be your infinite deck figure. Your final V is:


That concludes the initial way to estimate advantage, the next chapter 4, will detail estimates of expected value and optimal betting. Pappy’s excuse reffers to Pappy Smith, father of the owner, and the real founder of Harold’s Club, sadly missed in Reno (club and man). He more or less originated limiting doubling down to 10 or 11 only, to allegedly protect service men from throwing their money away doubling more, during WWII.

Chapter 4, the truly optimal bankroll, to infinity and beyond.

The World Blackjack Portal!