Transcript blackjack analysis

Blackjack: Myths vs. Reality
Group K
Andrew Kerr
Sven Skoog
Andrew Phillips
Woj Wrona
Agenda
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Blackjack: The Game
Assumptions
Simulation
Data Findings
Decision Tree
Descriptive Analysis
Question & Answers
Blackjack: The Game
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Each player is dealt two cards and is then offered the
opportunity to take more.
The hand with the highest total wins as long as it doesn't
exceed 21; a hand with a higher total than 21 is said to
bust. Cards 2 through 10 are worth their face value, and
face cards (Jack, Queen, King) are also worth 10.
An ace's value is 11 unless this would cause the player
to bust, in which case it is worth 1. A hand in which an
ace's value is counted as 11 is called a soft hand,
because it cannot be busted if the player draws another
card.
Assumptions
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$100 bet / hand
No splitting, double down, insurance
Samples with blackjack bonus and without bonus
Simulation
Percentage of Blackjack
Assumption: 10,000 Hands
Mean: 0.0454 or 4.5%
Percentage of Bust
Assumption: 10,000 Hands
Mean: 0.2803 or 28.03%
Normality of 1, 2 and 3 Card
Totals
Histogram of 3-Card-Sum
Normal
Mean 19.84
StDev 4.915
N
3333
300
Frequency
250
Histogram of 2-Card-Sum
Mean 14.47
StDev 4.121
N
5000
500
Frequency
8
12
16
20
3-Card-Sum
300
200
100
Normal
Mean 7.275
StDev 2.952
N
10000
3000
2500
Frequency
50
0
400
Histogram of 1 Card
150
100
Normal
Face card probability = 30%
200
0
6
9
12
15
2-Card-Sum
18
21
24
2000
1500
1000
500
0
2
4
6
8
10
12
14
Card
Approaching Normality as Card Count Increases
24
28
Why Dealer Stays at 17?
Dealer
Stays at..
Blackjack
16
17
18
4.7%
4.6%
4.9%
Bust
20.42%
28.23%
37.53%
Dealer
Beats 17
17.74%
27.85%
23.33%
Values derived from 10,000 card simulation
2 Card Total Confidence Interval
Probability of achieving a 2 card total 18 ≤ 2 cards ≤ 21 is 20.9%
3 Card Total Confidence Interval
Probability of achieving a 3 card total 18 ≤ 3 cards ≤ 21 is 37.1%
Regression of Aces vs. Wins

Do Aces result in Wins?
 Many players assume that an Ace increases the odds of winning
a hand

Regression model confirms that Aces lead to a very slight player
edge

Any edge is likely attributable to the flexibility of the Ace (1 or 11)
but not the fact that the Ace gives you an 11 ( no more powerful
that any other pairing, i.e. 8,3 or 7,4 that leads to an 11).
Regression of Aces vs. Wins
Two Card Play
Four Card Play
Three Card Play
Player=15, Dealer shows 7
A makes 16
7.7%
0
2 makes 17
7.7%
100
3 makes 18
7.7%
200
TRUE
Hit
0
0.0769
-100
A makes 16
0
0.0769
0
0.0769
2 makes 17
100
-30.79
4 makes 19
200
5 makes 20
7.7%
200
6 makes 21
7.7%
200
else BUST
53.9%
0
0.0769
3 makes 18
0.0769
-100
Stay
FALSE
100
4 makes 19
-100
0
7.7%
0
2 makes 17
3 makes 18
FALSE
-100
0
-200
7.7%
0
200
0
7.7%
0
400
Double
7.7%
200
-100
A makes 16
7.7%
200
5 makes 20
0
-100
0.0769
0
0.0769
100
-30.79
0.0769
-30.79
0.0769
Chance
100
Hit? (Card 3)
2-Cards = 15
7.7%
200
100
0.5386
7.7%
100
Chance
7.7%
7.7%
6 makes 21
7.7%
200
else BUST
53.9%
0
200
0.0769
100
0.0769
100
0.0769
100
0.5386
-100
Chance
-61.58
4 makes 19
7.7%
400
5 makes 20
7.7%
6 makes 21
7.7%
else BUST
53.9%
400
400
0
0
200
Win
Push
Lose
30.7%
7.7%
61.5%
0
200
0
200
0
-200
Player=17, Dealer shows 8
A makes 18
7.7%
100
2 makes 19
7.7%
3 makes 20
7.7%
200
200
4 makes 21
7.7%
200
5 makes 22
TRUE
Hit
0
0
A makes 18
0.077
100
2 makes 19
100
0.077
0
-100
3 makes 20
8 makes 25
9 makes 26
10 makes 27
200
7.7%
0.077
0
-100
7.7%
0.077
0
-100
7.7%
0.077
0
-100
7.7%
0.077
0
-100
30.8%
0.308
0
-100
4 makes 21
-46.16
FALSE
Stay
5 makes 22
0
7.7%
2 makes 19
7.7%
3 makes 20
7.7%
4 makes 21
7.7%
5 makes 22
7.7%
0.000
0
-200
6 makes 23
7.7%
0.000
0
-200
7 makes 24
7.7%
0.000
0
-200
7.7%
0.000
400
400
400
Take only one card
FALSE
-100
0.077
100
0.077
100
0.000
0
0.000
7 makes 24
200
0.000
200
0.000
8 makes 25
200
9 makes 26
Chance
-92.32
8 makes 25
0
-200
9 makes 26
7.7%
0.000
0
-200
10 makes 27
30.8%
0.000
0
-200
0.077
100
7.7%
0.077
0
-100
7.7%
0.077
0
-100
7.7%
0.077
0
-100
7.7%
0.077
0
-100
7.7%
0.077
0
-100
30.8%
0.308
0
-100
-46.16
-100
A makes 18
200
Double
0
Chance
6 makes 23
0
7.7%
200
Decision
-100
7.7%
0.077
-46.16
7 makes 24
2 cards = 17
7.7%
200
0.077
7.7%
7.7%
100
100
0.077
Chance
6 makes 23
Dealer 18
0.077
10 makes 27
Win
Push
Lose
23.0%
7.7%
69.2%
Player=11, Dealer shows 7
A makes 13
-100
7.7%
0.000
0
-100
3 makes 14
7.7%
0.000
0
-100
4 makes 15
7.7%
0.000
0
-100
7.7%
0.000
0
-100
6 makes 17
7.7%
0.000
7 makes 18
7.7%
5 makes 16
FALSE
0
0.000
0
2 makes 13
Hit
7.7%
Chance
A makes 12
2 makes 13
3 makes 14
15.39
100
200
8 makes 19
7.7%
9 makes 20
7.7%
200
200
10 makes 21
30.8%
200
0
0.000
4 makes 15
100
7.7%
0.077
0
-200
7.7%
0.077
0
-200
7.7%
0.077
0
-200
7.7%
0.077
0
-200
7.7%
0.077
0
-200
7.7%
0.077
0.000
100
0.000
5 makes 16
100
0.000
Chance
100
Decision
2 cards = 11
-100
30.79
30.79
FALSE
Stay
0
0
-100
A makes 13
7.7%
0
-200
2 makes 13
7.7%
0.077
0
-200
3 makes 14
7.7%
0.077
0
-200
4 makes 15
7.7%
0.077
0
-200
5 makes 16
7.7%
0.077
0
-200
6 makes 17
7.7%
0.077
7 makes 18
7.7%
8 makes 19
7.7%
9 makes 20
7.7%
Take only one card
Double
TRUE
-100
0.077
6 makes 17
200
7 makes 18
400
8 makes 19
200
400
400
400
10 makes 21
30.8%
400
9 makes 20
200
7.7%
400
0
0.077
0.077
7.7%
400
Chance
30.79
7.7%
10 makes 21
30.8%
400
200
0
0.077
200
0.077
200
0.077
200
0.308
200
0.077
200
0.308
Win
Push
Lose
53.8%
7.7%
38.4%
200
Player=15, Dealer shows 7
18
A makes 19
7.7%
200
2 makes 20
7.7%
200
FALSE
Hit
0
100
0
0
100
7.7%
100
7.7%
0.00591361
200
100
FALSE
76.9%
0
100
FALSE
Hit
0
-30.79
4 makes 20
7.7%
200
100
5 makes 21
7.7%
0.00591361
61.6%
0
-100
TRUE
100
100
FALSE
Hit
0
100
A makes 18
7.7%
0
200
100
2 makes 19
7.7%
0
200
100
19
A makes 20
200
-38.48
7.7%
0
200
100
7.7%
0
200
100
69.2%
FALSE
Hit
0
0
7.7%
0
200
100
2 makes 20
7.7%
0
200
100
0
84.6%
0
7.7%
7.7%
0
200
100
else BUST
76.9%
100
FALSE
Hit
0
-100
6 makes 21
7.7%
else BUST
53.9%
0
-100
FALSE
0
0
0
Double
Chance
92.3%
0
0
TRUE
-100
0.0769
100
else BUST
84.6%
17
A makes 18
0
2 makes 19
100
7.7%
0
200
100
FALSE
Hit
Chance
0
-84.62
Chance
-38.48
3 makes 20
FALSE
100
4 makes 21
else BUST
Hit? (Card 4)
0
0
Chance
-61.58
0
400
200
5 makes 20
7.7%
0
400
200
6 makes 21
7.7%
0
else BUST
53.9%
7.7%
53.9%
0
69.2%
0
0
7.7%
7.7%
200
0
0
7.7%
200
0.0769
-200
200
Stay
TRUE
100
200
0
-200
0.0769
100
0.5386
-100
0
100
0
-100
100
400
7.7%
200
0.0769
92.3%
0
100
100
-100
-100
7.7%
7.7%
200
0.5386
7.7%
200
100
200
0
0.0769
3 makes 18
0
6 makes 21
TRUE
7.7%
else BUST
7.7%
400
Hit? (Card 4)
2 makes 21
100
2 makes 17
4 makes 19
-84.62
else BUST
100
0
200
-100
FALSE
0
200
-69.24
-100
0
100
0
-100
-25.467751
Stay
200
FALSE
Hit? (Card 3)
2-Cards = 15
A makes 16
20
7.7%
Chance
0
200
7.7%
-100
Hit? (Card 4)
200
A makes 21
-30.79
Stay
100
else BUST
7.7%
Stay
100
-100
0.0769
A makes 21
0
200
Stay
TRUE
200
0
0.0769
0
100
Stay
5 makes 20
TRUE
100
0.04733195
0
Hit? (Card 4)
0
100
61.6%
Hit? (Card 4)
-53.86
3 makes 21
0
4 makes 19
Hit? (Card 4)
else BUST
100
0.00591361
Chance
200
200
Hit
FALSE
Hit
0
5 makes 21
7.7%
Chance
-69.24
else BUST
Stay
TRUE
0.00591361
200
0.0769
A makes 20
7.7%
0
A makes 19
7.7%
-25.467751
Chance
2 makes 21
-30.79
-100
0
200
0
100
200
100
Stay
TRUE
Hit
100
0.00591361
Chance
4 makes 20
Hit? (Card 4)
0
100
0
Chance
0
7.7%
7.7%
0
0.00591361
200
TRUE
Hit
0
3 makes 18
3 makes 19
-100
-100
else BUST
0
0.00591361
200
100
0.04733195
0
Stay
7.7%
7.7%
Hit? (Card 4)
0
0.0769
2 makes 18
0.00591361
0
4 makes 21
Hit? (Card 4)
7.7%
100
Chance
3 makes 20
7.7%
A makes 17
-30.79
0
7.7%
16
Hit? (Card 4)
Stay
2 makes 17
200
0.00591361
200
0
A makes 16
0
Stay
0
0.00591361
else BUST
200
TRUE
100
7.7%
200
-53.86
else BUST
TRUE
Hit
Chance
3 makes 21
7.7%
2 makes 18
3 makes 19
0
0
A makes 17
0.0769
0
0
100
0
100
0
-100
House Edge (and Sensitivity)
…many complicating factors…
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Play on Hunches (ex: “stop on 17”), no Split, no Double
‘Educated Hunches’ (modified basic strat.), Split + Double
Basic Strategy (perfect play), 8-decks, blackjack pays 3:2
Basic Strategy (perfect play), 1-deck, blackjack pays 3:2
5%-15%
3%-5%
0.5%
≈ 0%
(not typical)
Basic Strategy (perfect play), 1-deck, blackjack pays 6:5
≈ 1.4%
Basic Strategy plus variable-bet card counting
≈ 1.0%
(plus broken kneecaps)
How much difference does splitting/doubling-down make?
± 3.5%
How much difference does blackjack payoff make?
± 1.3%
How much difference does surrender make?
± 0.8%
How much difference from deviations/slip-ups?
1 slip/hr ≈ 1%
(basic strategy becomes hunch)
Two, Three, and Four Card Game
Payout Over Time
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
House has distinct advantage over the player as additional cards are
played.
Head to Head
2 card
3 card
4 card
Player
45.70%
40.47%
21.16%
Dealer
47.48%
55.90%
77.92%
Push
6.82%
3.63%
0.92%
The house edge results from the fact that the player acts first and risks
busting before the dealer.
Two, Three, and Four Card Game
Payout Over Time
The End