Blackjack: An Analysis of Probability

By: John Theobald

So, How Do I Play?

• Start with 2 cards: one face down, one face up.• Objective: Closest to 21 without going over.• Decks normally interchanged every hand or so

in order to prevent cheating• “Draw/Hit” to take one card• “Stand/Stay” to stop drawing cards; ends turn• 2 through 9 represented by self-values, all face

cards worth 10, aces worth 1 or 11.

Sounds Easy Enough… Right?

• Dealer wins all ties (including 21), but loses if he busts.

• Dealer always goes last, so all moves by players must be made before the dealer plays.

• At a casino, the odds are stacked against the players, due to these dealer rules.

• Casinos use multiple decks to prevent card counting.

Great… Then How Do I Win?

• Form a basic strategy.• Basic strategy - the strategy which maximizes the

player's average gain, or expectation, playing one hand against a complete pack of cards (Griffin 12)

• Assess the probabilities of your hand to the remaining cards and determine whether you should hit or stay.

• Bet smart: proper betting can be the difference between profit and loss. Higher bets compensate for lost hands, but variance in betting is key to not losing too much.

Strategy Variations

• Hard draw/stand: Never hit 17 or higher.• Soft draw/stand: Draw up to 17, stand on 18.

Hit on 18 only if against dealer with 9 or 10• “Mimic the Dealer”: mimic the dealer’s

strategy. Odds still in the dealers favor, but can be useful as strategy variation. Still worse than a basic strategy though.

Hit or Stand?

• 21: Stay; you win unless dealer also has 21• 18-20: Very unlikely to get 21, stay.• 15-17: Depends on strategy chosen.• 1-14: Hit or pair split• Use your basic strategy or use a combination of

probabilities/luck to determine your next action.

• Observe the dealer’s “up card” to help make a decision.

Dealer’s Chance to Bust

2 3 4 5 6 7 8 9 T A

35 38 40 43 42 26 24 23 21 11

Up Card

% Bust

Conditional Probability

• P(A|B) = Probability of A, knowing that B already occurred.

• P(A|B) = P(A and B)/P(B)• Example: What is the probability you are

dealt (T,A)? • P(21|T dealt) = P(21 and Ace dealt)/P(A) =

Multiple Decks vs. Single Decks

• Multiple decks slightly decrease the odds of obtaining 21. For example,

• One deck:

• Two decks:

Betting

• Betting is performed at the beginning of each round.

• Payouts for winning hands vary by casino. Winning odds are normally 3:2. ($2 bet, $3 payout).

• Bets should be high enough to make up for lost hands, yet low enough to not go into debt. Variance is vital.

Betting Probabilities• Measured in expectation (expected value) for

series of rounds.• E[X] = x1p1 + x2p2 + … + xkpk • Where x = units bet, p= probability of success• Losses have a negative output.• Var(X) = • σ = variance of a single card payoff• n = sample size of cards drawn without

replacement

Blackjack Theory: Gambler’s Ruin

• Theory in which one starts off winning, but end up going broke as the rounds progress.

• Allen Wilson came up with an equation dealing with this:

where r = ruin probability, , n = number of units bet• Problems: assumes constant win rate, does

not account for blackjack strategies increasing win rate

Sources

• http://www.bjmath.com/bjmath/probable/condit.htm

• Beat The Dealer, Edwin O. Thorp• The Theory of Blackjack, Peter Griffin• Blackjack Essays, Mason Malmuth• The Casino Gambler’s Guide, Allen Wilson