Blackjack: An Analysis of Probability

1. Blackjack: An Analysis of Probability By: John Theobald

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. Dealer’s Chance to Bust Up Card % Bust

8. 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) =

9. Multiple Decks vs. Single Decks • Multiple decks slightly decrease the odds of obtaining 21. For example, • One deck: • Two decks:

10. 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.

11. 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

12. 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

13. 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