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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.