Blackjack: A Beatable Game

David ParkerAdvisor: Dr. Wyels

California Lutheran University ‘05

Why is Blackjack Beatable? Only game in a casino where the probabilities

change from game to game.

If a player can take full advantage of favorable probabilities, they might be able to win more money then the dealer over a period of time.

Rules of Blackjack Player(s) vs. Dealer

Object: Closest to 21 without going over

Card Values

Face Cards = 10

Aces = 1 or 11 (Player’s choice)

2,3,4,5,6,7,8,9,10 = Numerical value of card drawn.

Rules of Blackjack

Player Dealer

2 3 4 5 6 7 8 9 10 A

5 H H H H H H H H H H 6 H H H H H H H H H H 7 H H H H H H H H H H 8 H H H H H H H H H H 9 H D D D D D H H H H

10 D D D D D D D D H H 11 D D D D D D D D D H 12 H H S S S S H H H H 13 S S S S S S H H H H 14 S S S S S S H H H H 15 S S S S S S H H H H 16 S S S S S S H H S H 17 S S S S S S S S S S 18 S S S S S S S S S S 19 S S S S S S S S S S 20 S S S S S S S S S S 21 S S S S S S S S S S

2 3 4 5 6 7 8 9 10 A

2-2 P P P P P P H H H H 3-3 P P P P P P H H H H 4-4 H H H H H H H H H H 5-5 D D D D D D D D H H 6-6 H P P P P P H H H H 7-7 P P P P P P H H H H 8-8 P P P P P P P P P P 9-9 P P P P P P P P S S

10-10 S S S S S S S S S S A-A P P P P P P P P P P A2 H H H D D H H H H H A3 H H H D D H H H H H A4 H H D D D H H H H H A5 H H D D D H H H H H A6 H D D D D H H H H H A7 S D D D D S S H H S A8 S S S S S S S S S S A9 S S S S S S S S S S

A10 S S S S S S S S S S

Basic Strategy

Pla

yer

Pla

yer

Dealer Card UpDealer Card Up

S = Stand

H = Hit

D = Double Down

P = Split Pair

How to Count Cards Dr. Edward Thorp (1962) High cards are good for the player. Card Counting

Cards 2,3,4,5,6 are worth +1 Cards 10,J,Q,K,A are worth -1 Cards 7,8,9 are neutral and are worth 0

Player keeps a running total of cards played in their head. Once the deck is reshuffled the count is reset to zero.

The Truecount

Julian H. Braun (1964) A high count becomes more beneficial to the player as

the number of cards played increases. A truecount of +8 after 8 cards have been played:

A truecount of +8 after 44 cards have been played:

456.044

20

00.18

8

Truecount (Cont.) Player still keeps track of count. Player keeps track of total number of cards

played. Complete Count = Count divided by the

number of decks have not been completely exhausted.

Truecount = Floor (Complete Count).

Maple Simulation

Dealer Card Up Player Cards Final Player Cards Outcome Count Probability of winning at

count Number of Cards Played Truecount Probability of Winning at

Truecount

1 Deck Shoe500 trials of 20,000 hands42

.21% 43

.75%

44.5

1%

45.6

7% 47.1

8%

47.9

4%

47.6

4%

47.8

6%

48.6

7%

49.3

4%

49.5

3%

52.2

3%

50.6

8%

49.9

9%

47.9

7%

47.8

1%

47.5

0%

47.2

8%

46.8

3%

45.9

1%

45.5

1%

49.1

7%

35%

40%

45%

50%

55%

≤-5 -4 -3 -2 -1 0 1 2 3 4 ≥5

True Count

Win

nin

g P

ecen

tag

e

Player Dealer

Count vs. Truecount (Player's Edge)

y = -2E-06x3 - 0.0002x2

+ 0.0082x - 0.0132

y = -1E-05x3 - 5E-05x2 + 0.0036x - 0.0172

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-10 -8 -6 -4 -2 0 2 4 6 8 10

Edge

Cou

nt

Count Truecount

6 Deck Shoe

Betting Strategies Thorp – Bet Count Braun – Bet Truecount Hi-Low

When the truecount is in the player’s favor (>2), bet 20 chips, otherwise bet 1 chip.

MIT Team Pick a betting unit. When there is a favorable truecount (>2), bet the

[truecount x (betting unit)]. Otherwise bet half the betting unit.

Maple Simulation Dealer Card Up Player Cards Final Player Cards Outcome Count Probability of winning at

count Number of Cards Played Truecount Probability of Winning at

Truecount

Betting Consistently Thorp Braun Hi-Low MIT Blackjack Team Amount Bet Amount Won/Lost Total amount Won/Lost

Maple Simulation (Cont.) Study was conducted with the same rules as if we

were playing at a 5 dollar minimum Las Vegas blackjack table.

6 deck shoe.

Single player vs. dealer.

Trials of 500 hands 500 hands takes between 7.5 – 10 human hours to play.

Normal Distributions10,000 trials of 500 hands

-400 -300 -200 -100 0 100 200 300 400

Number of Chips WonNot Counting Thorp Braun MIT Blackjack Team Hi-Low

-10.41

-5.870.55

-7.59 6.09

Max Wins and Losses10,000 Trials of 500 Hands

-859.5

-240.5

-746.5

-366

Not Counting Braun

Hi - Low

MIT Team

Thorp

-96

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Ch

ips

Max Amount Won10,000 Trials of 500 Hands

Not Counting

Braun

Thorp

HI - Low

MIT Team

0 20 40 60 80 100 120

Chips

95% Confidence Intervals

Conclusions

Normal Distributions

-400 -300 -200 -100 0 100 200 300 400

Number of Chips WonNot Counting Thorp Braun MIT Blackjack Team Hi-Low

6.09

Conclusions Hi-Low strategy wins the most money.

Chances of getting caught are high. High Standard Deviation. Need to buy 860 Chips.

Normal Distributions

-400 -300 -200 -100 0 100 200 300 400

Number of Chips WonNot Counting Thorp Braun MIT Blackjack Team Hi-Low

0.55

Conclusions Hi-Low strategy wins the most money.

Chances of getting caught are high. High Standard Deviation. 860 Chips to Play.

MIT Strategy is the only other strategy in which the player wins money Proven to work. Good Standard Deviation. 366 Chips to Play.

Conclusions Not many chips (0.55) earned for number of

hours spent playing (7-10 hours). Dealers are taught the betting strategies to

spot card counters. Casinos take measures to improve their odds.

Not allowing the player to double down with certain hands.

Dealer has to hit on 17. Reshuffling with cards left in the shoe.

However….

1 Deck Shoe500 trials of 20,000 hands

42.2

1%

43.7

5%

44.5

1%

45.6

7%

47.1

8%

47.9

4%

47.6

4%

47.8

6%

48.6

7%

49.3

4%

49.5

3%52.2

3%

50.6

8%

49.9

9%

47.9

7%

47.8

1%

47.5

0%

47.2

8%

46.8

3%

45.9

1%

45.5

1%

49.1

7%

35%

40%

45%

50%

55%

≤-5 -4 -3 -2 -1 0 1 2 3 4 ≥5

True Count

Win

nin

g P

ecen

tag

e

Player Dealer

Single Deck Blackjack

47.94 47.81

0

20

40

60

80

100

Player Dealer

• Player has a 0.13% edge on the dealer!• 0.0013*500 = 0.65• Better than all 6-deck strategies with the

exception of the Hi-Low Method.• Recommendation: learn basic strategy and find

a 1-deck game that reshuffles after every hand!

Further Studies Rules Variations

Player is allowed to re-split aces. Blackjack pays 6-5 instead of 2-1.

Play at numerous tables. Increase the number of players. Various other card counting strategies. Write an NSF grant to obtain funding to test

findings in a Casino setting.

References• Baldwin, Roger, Wilbert Cantey, Herbert Maisel, and James McDermott.

"The Optimum Strategy to Blackjack." Journal of the American Statistical Association 51.275 (1956): 429-439.

• Manson, A.R., A.J. Barr, and J.H. Goodnight. "Optimum Zero-Memory Strategy and Exact Probabilities for 4-deck Blackjack." The American Statistician May 1975: 84-88.

• Mezrich, Ben. Bringing Down the House. 1st ed. New York: Free Press, 2003.

• Millman, Martin. "A Statistical Analysis of Casino Blackjack." The American Mathematical Monthly Aug - Sep 1983: 431-436.

• Tamhane, Ajit, and Dorothy Dunlop. Statistics and Data Analysis. Upper Saddle River: Prentice Hall, 2000.

• Thorp, Edward. "A Favorable Strategy for twenty-one." Proc Natl Acad Sci Jan 1961: 110–112.

• Thorp, Edward. Beat the Dealer. 2nd ed. New York: Random House, 1966.

• Thorp, Edward. The Mathematics of Gambling. 1st ed. New York: Gambling Times, 1985.

• Larsen, Richard, and Morris Marx. An Introduction to Mathematical Statistics and its Applications. 2nd ed. Eaglewood Cliffs: Prentice Hall, 2000.

Special Thanks! Dr. Cindy Wyels – California Lutheran University.

Dr. Karrolyne Fogel – California Lutheran University.

Dr. David Kim – Manhattan College.

Larry Coaly – California Lutheran University.

Bryan Parker – University of California Los Angeles.