1. Craps Table Odds: Comparing Player’s Edges For Selected Table Odds
David Wirkkala
May 7, 2002
Advisor: Dr. Karrolyne Fogel

2. 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

3. The Craps Table

4. 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

5. 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

6. 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%

7. 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

8. 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

9. 1x Odds Empirical Results

10. Normal Distribution?

11. 2x Odds Empirical Results

12. 3, 4, 5x Odds Empirical Results

13. 5x Odds Empirical Results

14. 10x Odds Empirical Results

15. Average Winnings (dollars)
Empirical Results
Average Winnings (dollars)
Standard Deviation
(dollars)
Average Bet
Player’s Edge
(%)
1x odds
2x odds
345x odds
5x odds
10x odds

16. Empirical Results

17. Empirical Results

18. Empirical Results

19. Empirical Results

20. 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:

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

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

23. Empirical Player’s Edges vs. Theoretical Player’s Edges
Empirical Results
Theoretical Expectations
1x odds % %
2x odds % %
3, 4, 5x odds % %
5x odds % %
10x odds % %

24. Assessment Maple program was written correctly Random dice
Winnings appear to be normally distributed
10x Table Odds gives the player the best edge

25. 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

26. Resources Dr. Karrolyne Fogel Art Benjamin Anna P.
Probability and Statistical Inference
Hogg and Tanis
Applications of Discrete Mathematics
Rosen