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

Average Winnings (dollars)   Empirical Results   Average Winnings (dollars) Standard Deviation (dollars) Average Bet 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

Empirical Results

Empirical Results

Empirical Results

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:

Calculating The Theoretical Player’s Edge For 1x Table Odds Overall probability of winning: Overall probability of losing: Overall player’s average gain:

Calculating The Theoretical Player’s Edge For 1x Table Odds Player’s average gain: Player’s average bet: The player’s edge is:

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