Introduction and Tips

“Just another 2 bucks down the drain. You bastards! No. Calm down. Learn to enjoy losing.” – Hunter Thompson, Fear and Loathing in Las Vegas

Blackjack is unique among casino games due to the nature of the house’s odds – odds which can be won back by superior strategizing and execution of an optimal card counting system. Here we explore the method of a simple card counting system and how card counting can result in higher returns.

A Few Clarifications…

Card Counters Don’t Cheat!

• Card counting is not prohibited by any law.• Swinging the odds in your favor is part of the game and does not conflict with any rules of blackjack.• Casinos have no legal avenue to prosecute card counters (though they do have the right to bar any individual from entering the casino for any reason they see fit).

Card Counting Doesn’t Always Yield Profit

Those of you familiar with the popular book Bringing Down the House might recall a situation where it was probabilistically correct for the main character to make a large bet as the deck was “rich”, or contained an extremely favorable amount of high cards. Kevin lost about $50,000 on that deck, and he actually made the right play!

Right, so onto more interesting stuff…

How Card Counters Operate:

The goal of card counting is to keep track of cards which have already been played, thereby giving the card counter a better idea of what cards are left in the deck. The most efficient way card counters have found to remember cards is by assigning counting values. Each card is put into a group based on its playing value (ie 1,…,11), and then each group is assigned a counting value which the card counter keeps track of during play.

Example: Harvey Dubner System

2 3 4 5 6 7 8 9 10 J Q K A

Now, suppose we are at a table where the first hand has been dealt and the dealer has just revealed his down card. Given the following hands:

Dealer Eric Mike Dan* Peter (10,7) (4,8,6) (3,8,2,5) (9,3,Q) (A,9)

One can see that each hand adds up to its own total, and the running count of the hands above is

The easiest way to keep track of the running count is to mentally group the cards in such a way as to cancel them out:

*Note that Dan was the only player to lose his hand.

Card Counting in Blackjack

Nickolaus Groh ’07Swarthmore College, Department of Mathematics & Statistics

References

Griffin, Peter A. 1999. The Theory of Blackjack. Huntington Press, Las Vegas.

Larson, Richard J. and Morris L. Marx. 2001. An Introduction to Mathematical Statistics and Its Application. Prentice Hall, Upper Saddle River.

Mezrich, Ben. 2002. Bringing Down the House. Free Press, New York.

Smith, Ken. 1996. Rules of blackjack, free blackjack game, and basic strategy engine. http://www.blackjackinfo.com/. Accessed 12 September 2006.

Thorp, Edward O. and William E. Walden. 1973. The Fundamental Theorem of Card Counting with Applications to Trente-et-Quarante and Baccarat. International Journal of Game Theory. Vol. 2 Issue 2.

Tip 1:A crazy outfit is one of many good ways to fool the eyes in the sky.

AcknowledgementsI would like to thank Professor Walter Stromquist for his help with pdf’s,

Dan Sullivan for his undying patience, Sam Graffeo for being an extreme, Eric Zwick for being my inspiration, Peter Kriss for my Kantian notion of duty, and Michael Karcher for the reset button (because sometimes we need a different universe).

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The Expected Value Function:

Given X is the average of single card payoffs computed from a random sample of n cards, we have

where each Xi represents the next card which could be drawn from a population of cards whose (payoff) variance is σ2. Define the variance of X drawn without replacement as

Now, suppose our population of cards has a single card payoff average of μ. According to the central limit theorem, the distribution of X will tend towards normal distribution as the sample size increases. Thus, we have

Notice that we have defined μ < 0. In truth we could have μ ≤ 0 since we assume that we are starting with a full deck and have not counted any cards yet. Assume now that we have counted c cards, all of which had negative payoffs. Then we see a shift in fX(x) as follows:

This shift reflects the absence of below average payoff cards which we have counted, which effectively increases μ. It is also important to mention that due to the removal of these c cards, our new pdf fX(x)* should have less variance and be slightly “taller” than our old pdf fX(x).

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Tip 2:Try not to get caught – digits are useful.

… Now, X is a continuous random variable with the pdf fX(x). Thus, the expected value of X is defined as

So, given our normally distributed pdf, a card counter’s expectation with n – c cards remaining can be approximated by the integral

Tip 3:Read this book.

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