Blackjack: Myths vs. Reality

Group KAndrew Kerr Andrew PhillipsSven Skoog Woj Wrona

Agenda

Blackjack: The Game Assumptions Simulation Data Findings Decision Tree Descriptive Analysis Question & Answers

Blackjack: The Game

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

$100 bet / hand No splitting, double down, insurance Samples with blackjack bonus and without bonus

Simulation

Percentage of Blackjack

Mean: 0.0454 or 4.5%Assumption: 10,000 Hands

00

0 1 2

0% 4.54% 0 0

Mean=0.0454

Distribution of Blackjack

0

5

10

15

20

25

Mean=0.0454

0 1 2

@RISK Student VersionFor Academic Use Only

Percentage of Bust

Mean: 0.2803 or 28.03%Assumption: 10,000 Hands

1111

0 0.5 1 1.5 2

95% 0% 0 1

Mean=0.2803 Mean=0.2803 Mean=0.2803

Distribution of Bust

0

2

4

6

8

10

12

14

16

Mean=0.2803

0 0.5 1 1.5 2

@RISK Student VersionFor Academic Use Only

Card

Fre

qu

en

cy

1412108642

3000

2500

2000

1500

1000

500

0

Mean 7.275StDev 2.952N 10000

Normal Histogram of 1 Card

Normality of 1, 2 and 3 CardTotals

Approaching Normality as Card Count Increases

Face card probability = 30%3-Card-Sum

Fre

qu

en

cy

28242016128

300

250

200

150

100

50

0

Mean 19.84StDev 4.915N 3333

Histogram of 3-Card-SumNormal

2-Card-Sum

Fre

qu

en

cy

242118151296

500

400

300

200

100

0

Mean 14.47StDev 4.121N 5000

Histogram of 2-Card-SumNormal

Why Dealer Stays at 17?

Dealer Stays at..

16 17 18

Blackjack 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

2121

4 10 16 22

79.1% 20.9% 0% 18 21

Mean=14.5539

Probability of win w/ 2 card

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

Mean=14.5539

4 10 16 22

@RISK Student VersionFor Academic Use Only

Probability of achieving a 2 card total 18 ≤ 2 cards ≤ 21 is 20.9%

3 Card Total Confidence Interval

18181818

5 10 15 20 25 30

45.6% 37.1% 17.3% 18 21

Mean=18.6981 Mean=18.6981

Probability of win w/ 3 card

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

0.180

0.200

Mean=18.6981

5 10 15 20 25 30

@RISK Student VersionFor Academic Use Only

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

Three Card Play

Four Card Play

Player=15, Dealer shows 77.7% 0.0769

0 -100

7.7% 0.0769

100 0

7.7% 0.0769

200 100

TRUE Chance

0 -30.79

7.7% 0.0769

200 100

7.7% 0.0769

200 100

7.7% 0.0769

200 100

53.9% 0.5386

0 -100

Hit? (Card 3)

-100 -30.79

FALSE 0

0 -100

7.7% 0

0 -200

7.7% 0

200 0

7.7% 0

400 200

FALSE Chance

-100 -61.58

7.7% 0

400 200

7.7% 0

400 200

7.7% 0

400 200

53.9% 0

0 -200

2-Cards = 15

Hit

Stay

Double

A makes 16

2 makes 17

3 makes 18

4 makes 19

5 makes 20

6 makes 21

else BUST

A makes 16

2 makes 17

3 makes 18

4 makes 19

5 makes 20

6 makes 21

else BUST

7.7% 0.0769

0 -100

7.7% 0.0769

100 0

7.7% 0.0769

200 100

Chance

-30.79

7.7% 0.0769

200 100

7.7% 0.0769

200 100

7.7% 0.0769

200 100

53.9% 0.5386

0 -100

A makes 16

2 makes 17

3 makes 18

4 makes 19

5 makes 20

6 makes 21

else BUST

Win Push Lose

30.7% 7.7% 61.5%

Player=17, Dealer shows 87.7% 0.077

100 0

7.7% 0.077

200 100

7.7% 0.077

200 100

7.7% 0.077

200 100

7.7% 0.077

0 -100

TRUE Chance

0 -46.16

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

Decision

-100 -46.16

Dealer 18 FALSE 0

0 -100

7.7% 0.000

200 0

7.7% 0.000

400 200

7.7% 0.000

400 200

7.7% 0.000

400 200

7.7% 0.000

Take only one card 0 -200

FALSE Chance

-100 -92.32

7.7% 0.000

0 -200

7.7% 0.000

0 -200

7.7% 0.000

0 -200

7.7% 0.000

0 -200

30.8% 0.000

0 -200

2 cards = 17

Hit

Stay

Double

A makes 18

2 makes 19

3 makes 20

4 makes 21

5 makes 22

6 makes 23

7 makes 24

8 makes 25

9 makes 26

10 makes 27

A makes 18

2 makes 19

3 makes 20

4 makes 21

5 makes 22

6 makes 23

7 makes 24

8 makes 25

9 makes 26

10 makes 27

7.7% 0.077

100 0

7.7% 0.077

200 100

7.7% 0.077

200 100

7.7% 0.077

200 100

7.7% 0.077

0 -100

Chance

-46.16

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

A makes 18

2 makes 19

3 makes 20

4 makes 21

5 makes 22

6 makes 23

7 makes 24

8 makes 25

9 makes 26

10 makes 27

Win Push Lose

23.0% 7.7% 69.2%

Player=11, Dealer shows 77.7% 0.000

0 -100

7.7% 0.000

0 -100

7.7% 0.000

0 -100

7.7% 0.000

0 -100

7.7% 0.000

0 -100

FALSE Chance

0 15.39

7.7% 0.000

100 0

7.7% 0.000

200 100

7.7% 0.000

200 100

7.7% 0.000

200 100

30.8% 0.000

200 100

Decision

-100 30.79

FALSE 0

0 -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

Take only one card 0 -200

TRUE Chance

-100 30.79

7.7% 0.077

200 0

7.7% 0.077

400 200

7.7% 0.077

400 200

7.7% 0.077

400 200

30.8% 0.308

400 200

2 cards = 11

Hit

Stay

Double

A makes 13

2 makes 13

3 makes 14

4 makes 15

5 makes 16

6 makes 17

7 makes 18

8 makes 19

9 makes 20

10 makes 21

A makes 13

2 makes 13

3 makes 14

4 makes 15

5 makes 16

6 makes 17

7 makes 18

8 makes 19

9 makes 20

10 makes 21

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

Chance

30.79

7.7% 0.077

200 0

7.7% 0.077

400 200

7.7% 0.077

400 200

7.7% 0.077

400 200

30.8% 0.308

400 200

A makes 12

2 makes 13

3 makes 14

4 makes 15

5 makes 16

6 makes 17

7 makes 18

8 makes 19

9 makes 20

10 makes 21

Win Push Lose

53.8% 7.7% 38.4%

Player=15, Dealer shows 77.7% 0.00591361

100 0

7.7% 0.00591361

200 100

7.7% 0.00591361

200 100

TRUE Chance

0 -30.79

7.7% 0.00591361

200 100

7.7% 0.00591361

200 100

61.6% 0.04733195

0 -100

Hit? (Card 4)

-30.79

FALSE 0

0 -100

Hit

Stay

A makes 17

2 makes 18

3 makes 19

4 makes 20

5 makes 21

else BUST

7.7% 0

200 100

7.7% 0

200 100

FALSE Chance

0 -38.48

7.7% 0

200 100

7.7% 0

200 100

69.2% 0

0 -100

Hit? (Card 4)

0

TRUE 0.0769

100 0

Hit

Stay

A makes 18

2 makes 19

3 makes 20

4 makes 21

else BUST

7.7% 0

200 100

7.7% 0

200 100

FALSE Chance

0 -53.86

7.7% 0

200 100

76.9% 0

0 -100

Hit? (Card 4)

100

TRUE 0.0769

200 100

Hit

Stay

A makes 19

2 makes 20

3 makes 21

else BUST

7.7% 0

200 100

FALSE Chance

0 -69.24

7.7% 0

200 100

84.6% 0

0 -100

Hit? (Card 4)

100

TRUE 0.0769

200 100

Hit

Stay

A makes 20

2 makes 21

else BUST

7.7% 0

200 100

FALSE Chance

0 -84.62

92.3% 0

0 -100

Hit? (Card 4)

100

TRUE 0.0769

200 100

Hit

Stay

A makes 21

else BUST

7.7% 0.0769

200 100

53.9% 0.5386

0 -100

6 makes 21

else BUST

16

17

18

19

20

7.7% 0.00591361

100 0

7.7% 0.00591361

200 100

7.7% 0.00591361

200 100

TRUE Chance

0 -30.79

7.7% 0.00591361

200 100

7.7% 0.00591361

200 100

61.6% 0.04733195

0 -100

7.7% Hit? (Card 4)

0 -30.79

FALSE 0

0 -100

7.7% 0

200 100

7.7% 0

200 100

FALSE Chance

0 -38.48

7.7% 0

200 100

7.7% 0

200 100

69.2% 0

0 -100

7.7% Hit? (Card 4)

0 0

TRUE 0.0769

100 0

7.7% 0

200 100

7.7% 0

200 100

FALSE Chance

0 -53.86

7.7% 0

200 100

76.9% 0

0 -100

7.7% Hit? (Card 4)

0 100

TRUE 0.0769

200 100

TRUE Chance

0 -25.467751

7.7% 0

200 100

FALSE Chance

0 -69.24

7.7% 0

200 100

84.6% 0

0 -100

7.7% Hit? (Card 4)

0 100

TRUE 0.0769

200 100

7.7% 0

200 100

FALSE Chance

0 -84.62

92.3% 0

0 -100

7.7% Hit? (Card 4)

0 100

TRUE 0.0769

200 100

7.7% 0.0769

200 100

53.9% 0.5386

0 -100

Hit? (Card 3)

-100 -25.467751

FALSE 0

0 -100

7.7% 0

0 -200

7.7% 0

200 0

7.7% 0

400 200

FALSE Chance

-100 -61.58

7.7% 0

400 200

7.7% 0

400 200

7.7% 0

400 200

53.9% 0

0 -200

2-Cards = 15

Hit

Stay

Double

A makes 16

2 makes 17

3 makes 18

4 makes 19

5 makes 20

6 makes 21

else BUST

A makes 16

2 makes 17

3 makes 18

4 makes 19

5 makes 20

6 makes 21

else BUST

Hit

Stay

A makes 21

else BUST

Hit

Stay

A makes 20

2 makes 21

else BUST

Hit

Stay

A makes 19

2 makes 20

3 makes 21

else BUST

Hit

Stay

A makes 18

2 makes 19

3 makes 20

4 makes 21

else BUST

Hit

Stay

A makes 17

2 makes 18

3 makes 19

4 makes 20

5 makes 21

else BUST

House Edge (and Sensitivity)…many complicating factors…

Play on Hunches (ex: “stop on 17”), no Split, no Double 5%-15% ‘Educated Hunches’ (modified basic strat.), Split + Double 3%-5% Basic Strategy (perfect play), 8-decks, blackjack pays 3:2 0.5% Basic Strategy (perfect play), 1-deck, blackjack pays 3:2 ≈ 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

House has distinct advantage over the player as additional cards are played.

The house edge results from the fact that the player acts first and risks busting before the dealer.

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%

Two, Three, and Four Card Game Payout Over Time

The End