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

1x odds 2x odds 345x odds 5x odds 10x odds Average Winnings (dollars) -2.1214404

Standard Deviation (dollars) 23.17060673

Average Bet (dollars) 250.003398

Player’s Edge (%) -0.848564626

-2.120945

-2.10254

37.18120493

60.19374956

370.838377

566.671566

-0.571932446

-0.371033263

-2.147962

-2.181114

71.3223923

132.3755179

650.003398

-0.330453965

1150.003398

-0.189661527

-2.09

-2.1

-2.11

-2.12

-2.13

-2.14

-2.15

-2.16

-2.17

-2.18

-2.19

Empirical Results

Average Winnings After 100 Games

345x odds 1x odds 2x odds 5x odds 10x odds

1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 Empirical Results

Average Bet After 100 Games

10x odds 1x odds 2x odds 345x odds 5x odds

Empirical Results 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

Standard Deviation After 100 Games

10x odds 1x odds 2x odds 345x odds 5x odds

0 -0.1

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

-0.9

Empirical Results

Player's Edge

345x odds 5x odds 10x odds 2x odds 1x odds

Calculating The Theoretical Player’s Edge For 1x Table Odds • Probability of player winning on Comeout Roll:

pr(7)



pr(11)



6 36



2 36



8 36

• Probability of player establishing a point and then winning:

pr(

4 ) *

pr

( 4

before

7 ) 

pr(

8 ) *

pr

( 8

before

7 ) 

pr(

5 ) *

pr

( 5

before

7 ) 

pr(

9 ) *

pr

( 9

before

7 ) 

pr(

6 ) *

pr

( 6

before

7 ) 

pr(

10 ) *

pr

( 10

before

7 ) or    3 36       3 9        4 36       4 10        5 36       5 11        5 36       5 11        4 36       4 10        3 36       3 9     9648 35640 • Overall probability of player winning: 8 36  9648 35640  244 495

Calculating The Theoretical Player’s Edge For 1x Table Odds • Overall probability of winning: 244 495 • Overall probability of losing: 1  244 495  251 495 • Overall player’s average gain: 244 495  251 495   7 495

Calculating The Theoretical Player’s Edge For 1x Table Odds • Player’s average gain:  7 495 • Player’s average bet: 1     3 36        4 36        5 36        5 36        4 36        3 36       5 3 • The player’s edge is: 7 495 5 3   .

848 %

Empirical Player’s Edges vs. Theoretical Player’s Edges 1x odds 2x odds 3, 4, 5x odds 5x odds 10x odds Empirical Results -0.84856 % -0.57193 % -0.37103 % -0.33045 % -0.18966 % Theoretical Expectations -0.848 % -0.572 % -0.374 % -0.326 % -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