Craps Table Odds: Comparing Player’s Edges For Selected Table Odds David Wirkkala May 7, 2002...
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Transcript of Craps Table Odds: Comparing Player’s Edges For Selected Table Odds David Wirkkala May 7, 2002...
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