Blackjack: A Beatable Game
David Parker
Advisor: 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
S = Stand
Basic Strategy
H = Hit
P = Split Pair
Dealer Card Up
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
3
H
H
H
H
D
D
D
H
S
S
S
S
S
S
S
S
S
4
H
H
H
H
D
D
D
S
S
S
S
S
S
S
S
S
S
5
H
H
H
H
D
D
D
S
S
S
S
S
S
S
S
S
S
6
H
H
H
H
D
D
D
S
S
S
S
S
S
S
S
S
S
7
H
H
H
H
D
D
D
S
S
S
S
S
S
S
S
S
S
8
H
H
H
H
H
D
D
H
H
H
H
H
S
S
S
S
S
9
H
H
H
H
H
D
D
H
H
H
H
H
S
S
S
S
S
10
H
H
H
H
H
H
D
H
H
H
H
S
S
S
S
S
S
A
H
H
H
H
H
H
H
H
H
H
H
H
S
S
S
S
S
Player
Player
Dealer Card Up
2
H
H
H
H
H
D
D
H
S
S
S
S
S
S
S
S
S
D = Double Down
2-2
3-3
4-4
5-5
6-6
7-7
8-8
9-9
10-10
A-A
A2
A3
A4
A5
A6
A7
A8
A9
A10
2
P
P
H
D
H
P
P
P
S
P
H
H
H
H
H
S
S
S
S
3
P
P
H
D
P
P
P
P
S
P
H
H
H
H
D
D
S
S
S
4
P
P
H
D
P
P
P
P
S
P
H
H
D
D
D
D
S
S
S
5
P
P
H
D
P
P
P
P
S
P
D
D
D
D
D
D
S
S
S
6
P
P
H
D
P
P
P
P
S
P
D
D
D
D
D
D
S
S
S
7
P
P
H
D
P
P
P
P
S
P
H
H
H
H
H
S
S
S
S
8
H
H
H
D
H
H
P
P
S
P
H
H
H
H
H
S
S
S
S
9
H
H
H
D
H
H
P
P
S
P
H
H
H
H
H
H
S
S
S
10
H
H
H
H
H
H
P
S
S
P
H
H
H
H
H
H
S
S
S
A
H
H
H
H
H
H
P
S
S
P
H
H
H
H
H
S
S
S
S
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:
20
 0.456
44

A truecount of +8 after 44 cards have been played:
8
 1.00
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
45%
≤-5
-4
50.68%
-3
-2
-1
Player
0
1
True Count
Dealer
2
3
4
45.51%
49.53%
45.91%
49.34%
46.83%
48.67%
47.28%
47.86%
47.50%
47.64%
47.81%
47.94%
47.97%
47.18%
49.17%
49.99%
50%
45.67%
44.51%
43.75%
42.21%
55%
52.23%
Winning Pecentage
1 Deck Shoe
500 trials of 20,000 hands
40%
35%
≥5
Count vs. Truecount (Player's Edge)
6 Deck Shoe
0.06
0.04
y = -2E-06x 3 - 0.0002x 2
+ 0.0082x - 0.0132
0.02
Count
0
-10
-8
-6
-4
-2
-0.02 0
-0.04
-0.06
-0.08
-0.1
-0.12
Edge
Count
Truecount
2
4
6
8
3
10
y = -1E-05x - 5E-05x 2 +
0.0036x - 0.0172
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 Distributions
10,000 trials of 500 hands
-10.41
0.55
-5.87
-7.59
-400
-300
Not Counting
-200
-100
6.09
0
100
200
Number of Chips Won
Thorp
Braun
MIT Blackjack Team
300
400
Hi-Low
Max Wins and Losses
10,000 Trials of 500 Hands
1000
Hi - Low
800
Thorp
600
Chips
400
200
MIT Team
Not Counting
Braun
-96
-240.5
0
-200
-400
-600
-800
-1000
-859.5
-366
-746.5
Max Amount Won
10,000 Trials of 500 Hands
MIT Team
HI - Low
Braun
Thorp
Not Counting
0
20
40
60
80
Chips
95% Confidence Intervals
100
120
Conclusions
Normal Distributions
6.09
-400
-300
Not Counting
-200
-100
0
100
200
Number of Chips Won
Thorp
Braun
MIT Blackjack Team
300
400
Hi-Low
Conclusions

Hi-Low strategy wins the most money.



Chances of getting caught are high.
High Standard Deviation.
Need to buy 860 Chips.
Normal Distributions
0.55
-400
-300
Not Counting
-200
-100
0
100
200
Number of Chips Won
Thorp
Braun
MIT Blackjack Team
300
400
Hi-Low
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….
100
Single Deck Blackjack
80
49.53%
60
45.51%
49.34%
45.91%
48.67%
46.83%
47.86%
47.28%
47.64%
47.50%
47.94%
47.81%
47.97%
49.17%
47.18%
49.99%
45.67%
44.51%
45%
43.75%
50%
42.21%
Winning Pecentage
55%
50.68%
52.23%
1 Deck Shoe
500 trials of 20,000 hands
40%
47.94
47.81
40
20
35%
≤-5
-4
-3
-2
-1
0
1
True Count
Player
Dealer
2
3
4
≥5
0
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.