Craps Table Odds:Comparing Player’s Edges For

Selected Table Odds

David Wirkkala

May 7, 2002

Advisor: Dr. Karrolyne Fogel

Dice Probability

In rolling two dice there are 36 possible outcomes.

• P(sum is 2) = P(sum is 12) = 1/36• P(sum is 3) = P(sum is 11) = 2/36• P(sum is 4) = P(sum is 10) = 3/36• P(sum is 5) = P(sum is 9) = 4/36• P(sum is 6) = P(sum is 8) = 5/36• P(sum is 7) = 6/36

The Craps Table

Betting Strategy: Conservative Craps

• Comeout Roll– Pass Line Bet ($1)

• Win on 7 or 11; pays 1:1 • Lose on 2, 3, or 12; replace bet and repeat Comeout Roll• If 4, 5, 6, 8, 9, or 10 is rolled this number becomes the “point”;

proceed to Point Roll

• Point Roll– True Odds Bet ($ amount determined by Table Odds)

• Win if “point” is rolled before rolling a 7; lose both bets on 7.• Pass Line Bet pays 1:1• True Odds Bet pays 2:1 if point is 4 or 10, 3:2 if point is 5 or 9,

and 6:5 if point is 6 or 8; house has no advantage

What Are Table Odds?

• 1x odds – The True Odds Bet can be 1x the Pass Line Bet

• 2x odds – The True Odds Bet can be 2x the Pass Line Bet– If the point is 6 or 8 the True Odds Bet can be 5/2x the Pass

Line Bet

• 3, 4, 5x odds– The True Odds Bet can be 3x the Pass Line Bet if the point

is 4 or 10, 4x if the point is 5 or 9, and 5x if the point is 6 or 8.

• 5x odds – The True Odds Bet can be 5x the Pass Line Bet

• 10x odds – The True Odds Bet can be 10x the Pass Line Bet

Player’s Edge

• The player’s edge is the player’s average gain divided by the player’s average bet.

• Example:– Bet $11 to win $10– 50% chance of winning– Average gain: ½ * (-11) + ½ * 10 = -0.5– Player’s edge: -0.5/11 = -1/22 or –4.454%

Project Objectives• Use Maple to simulate Conservative Craps

for selected Table Odds• Analyze empirical data• Compare empirical data with theoretical

expectations• Determine the Table Odds that gives the

best player’s edge• Consider extensions of project

Simulation

• Recorded winnings and the amount bet after a 100 game session– A game consists of the Comeout Roll,

establishing a point, and either rolling the point or rolling a 7.

• Simulated 500,000 sessions for each Table Odds

1x Odds Empirical Results

Normal Distribution?

2x Odds Empirical Results

3, 4, 5x Odds Empirical Results

5x Odds Empirical Results

10x Odds Empirical Results

Empirical Results

 

 Average Winnings

(dollars)

 Standard Deviation

(dollars)

 Average Bet

(dollars)

 Player’s Edge

(%)

 1x odds -2.1214404 23.17060673 250.003398 -0.848564626

 2x odds -2.120945 37.18120493 370.838377 -0.571932446

 345x odds -2.10254 60.19374956 566.671566 -0.371033263

 5x odds -2.147962 71.3223923 650.003398 -0.330453965

 10x odds -2.181114 132.3755179 1150.003398 -0.189661527

Empirical Results

Average Winnings After 100 Games

1x odds 2x odds

345x odds

5x odds

10x odds

-2.19

-2.18

-2.17

-2.16

-2.15

-2.14

-2.13

-2.12

-2.11

-2.1

-2.09

Ave

rag

e W

inn

ing

s (d

oll

ars

)

Empirical Results

Average Bet After 100 Games

1x odds

2x odds

345x odds5x odds

10x odds

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

Ave

rag

e B

ets

(do

llar

s)

Empirical Results

Standard Deviation After 100 Games

1x odds

2x odds

345x odds

5x odds

10x odds

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

Sta

nd

ard

Devia

tio

n (

do

llars

)

Empirical Results

Player's Edge

1x odds

2x odds

345x odds5x odds

10x odds

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

(%)

Calculating The Theoretical Player’s Edge For 1x Table Odds

• Probability of player winning on Comeout Roll:

• Probability of player establishing a point and then winning:

or

• Overall probability of player winning:

36

8

36

2

36

6pr(11)pr(7)

)76(*)6)75(*)5)74(*)4 beforeprpr(beforeprpr(beforeprpr(

)710(*)10)79(*)9)78(*)8 beforeprpr(beforeprpr(beforeprpr(

35640

9648

9

3

36

3

10

4

36

4

11

5

36

5

11

5

36

5

10

4

36

4

9

3

36

3

495

244

35640

9648

36

8

Calculating The Theoretical Player’s Edge For 1x Table Odds

• Overall probability of winning:

• Overall probability of losing:

• Overall player’s average gain:

495

244

495

251

495

2441

495

71

495

2511

495

244

Calculating The Theoretical Player’s Edge For 1x Table Odds

• Player’s average gain:

• Player’s average bet:

• The player’s edge is:

495

7

3

51

36

31

36

41

36

51

36

51

36

41

36

31

%848.

3

5495

7

Empirical Player’s Edges vs. Theoretical Player’s Edges

  

 

Empirical Results Theoretical Expectations

1x odds -0.84856 % -0.848 %

2x odds -0.57193 % -0.572 %

3, 4, 5x odds -0.37103 % -0.374 %

5x odds -0.33045 % -0.326 %

10x odds -0.18966 % -0.184 %

 

Assessment

• Maple program was written correctly

• Random dice

• Winnings appear to be normally distributed

• 10x Table Odds gives the player the best edge

Project Extensions

• Consider a different betting strategy for selected Table Odds

• Compare different betting strategies with the same Table Odds

• Consider win/loss limits

• Further investigate the distribution of winnings

Resources

• Dr. Karrolyne Fogel• Art Benjamin• Anna P.• www.thewizardofodds.com• Probability and Statistical Inference

Hogg and Tanis• www.statsoftinc.com/textbook/stdisfit.html• Applications of Discrete Mathematics

Rosen