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
Up Card
% Bust
2
3
4
5
6
7
8
9
T
A
35
38
40
43
42
26
24
23
21
11
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)
=
4 16
∗
52 51
4
52
=
16
51
Multiple Decks vs. Single Decks
• Multiple decks slightly decrease the odds of
obtaining 21. For example,
• One deck:
𝑃 𝐵𝑙𝑎𝑐𝑘𝑗𝑎𝑐𝑘 = 𝑃 𝐴 ∗ 𝑃 𝑇 𝐴 + 𝑃 𝑇 ∗ 𝑃 𝐴 𝑇
4
16 16 4
=
∗
+
∗
= .04827
52 51 52 51
• Two decks:
𝑃 𝐵𝑙𝑎𝑐𝑘𝑗𝑎𝑐𝑘 = 𝑃 𝐴 ∗ 𝑃 𝑇 𝐴 + 𝑃 𝑇 ∗ 𝑃 𝐴 𝑇 =
32
103
+
32
104
∗
8
103
= .04780
8
104
∗
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 2 52−𝑛
𝑛
51
• 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
1
with this: 𝑟 =
𝑛
1+𝑠
where r = ruin probability, s =
p
,
q
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/co
ndit.htm
• Beat The Dealer, Edwin O. Thorp
• The Theory of Blackjack, Peter Griffin
• Blackjack Essays, Mason Malmuth
• The Casino Gambler’s Guide, Allen Wilson