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Ask the Wizard #192
Matt from New Britain
A simple way to estimate the probabiity of winning is to assume that every card has a 12/13 probability of not matching the stated rank. To win this bet, the victim would have to do this successfully 52 times. The probability of 52 wins is (12/13)^{52} = 1.56%. A fair price to pay would be 63.2 to 1. At 20 to 1 Vince had a 67.3% advantage (ouch!).
Accoring to G.M., who is a better mathematician than I am, the actual probabiliy is 1.6232727%. The reason for the difference is the outcome of each pick is positively correlated to previous picks.
Dave K. from Ohio
Follow this link.
James from Chicago
The house edge of this game is 1.30% or 1.33%, as shown in my survey of Las Vegas blackjack rules, depending on whether the number of decks is five or eight. The odds are better in ANY game where blackjack pays 3 to 2. If you were to play this game, which you shouldn’t, you should still always stand on blackjack. Personally, I think the "World’s Most Liberal Blackjack" claim on the marquee is false advertising.
Scott from San Diego
Ideally, you should challenge the hand before it is over, while it is still easy to run back the cards. It doesn’t hurt to ask later than that, but you are not entitled to anything. This is getting outside my area of expertise, but the decision whether or not to review the tape would likely depend on the amount of money involved and your value as a player.
William R. from Las Vegas
I asked Barney Vinson this question, author of Ask Barney: An Insider’s Guide to Las Vegas. He said the casino would likely only rate one of the bets, in your case $25. An advantage to doing that is that it certainly lowers risk. This might be a good play if you needed to put in a lot of action, for example to qualify for an event you were invited to, and didn’t have much money to lose. However I think that if large bets were involved ($100 or over) it would set off a red flag, and you probably wouldn’t be invited to the next event.
Jim from Brick, NJ
I hope you’re happy, I added a new page to answer this question. Please see my Blackjack Appendix 19.
The question I have about this bet is that 14.41% still isn’t "statistically significant" [ i.e. p < 0.05 ] , which is usually taken to mean greater than two Standard Deviations from the Mean  or a probability of less than a *combined* 5% of the event happening randomly on EITHER end of the series.
How many Sevens would have to be rolled in 500 rolls before you could say that there is a less than 2.5% chance that the outcome was entirely random (i.e. that the outcome was statistically significant) ?
Many Thanks & BTW , yours is ABSOLUTELY the BEST web site on the subject of gambling odds & probabilities that I’ve found .... keep up the good work !!!
Plexus from Warwick, Rhode Island
Thank you for the kind words. You should not state the probability that the throws were nonrandom is p. The way it should be phrased is the probability that a random game would produce such a result is p. Nobody expected 500 rolls to prove or disprove anything. It wasn’t I who set the line at 79.5 sevens, but I doubt it was chosen to be statistically significant; but rather, I suspect the it was a point at which both parties would agree to the bet.
The 2.5% level of significance is 1.96 standard deviations from expectations. This can be found with the formula =normsinv(0.025) in Excel. The standard deviation of 500 rolls is sqr(500*(1/6)*(5/6)) = 8.333. So 1.96 standard deviations is 1.96 * 8.333 = 16.333 rolls south of expectations. The expected number of sevens in 500 throws is 500*(1/6) = 83.333. So 1.96 standard deviations south of that is 83.333 − 16.333 = 67. Checking this using the binomial distribution, the exact probability of 67 or fewer sevens is 2.627%.
Ian F. from Provo
Assuming the player always holds the most represented number, the average is 11.09. Here is a table showing the distribution of the number of rolls over a random simulation of 82.6 million trials.
Yahtzee Experiment
Rolls  Occurences  Probability 
1  63908  0.00077371 
2  977954  0.0118396 
3  2758635  0.0333975 
4  4504806  0.0545376 
5  5776444  0.0699327 
6  6491538  0.0785901 
7  6727992  0.0814527 
8  6601612  0.0799227 
9  6246388  0.0756221 
10  5741778  0.0695131 
11  5174553  0.0626459 
12  4591986  0.0555931 
13  4022755  0.0487016 
14  3492745  0.042285 
15  3008766  0.0364257 
16  2577969  0.0312103 
17  2193272  0.0265529 
18  1864107  0.0225679 
19  1575763  0.019077 
20  1329971  0.0161013 
21  1118788  0.0135446 
22  940519  0.0113864 
23  791107  0.00957757 
24  661672  0.00801056 
25  554937  0.00671837 
26  463901  0.00561624 
27  387339  0.00468933 
28  324079  0.00392347 
29  271321  0.00328476 
30  225978  0.00273581 
31  189012  0.00228828 
32  157709  0.00190931 
33  131845  0.00159619 
34  109592  0.00132678 
35  91327  0.00110565 
36  76216  0.00092271 
37  63433  0.00076795 
38  52786  0.00063906 
39  44122  0.00053417 
40  36785  0.00044534 
41  30834  0.00037329 
42  25494  0.00030864 
43  21170  0.0002563 
44  17767  0.0002151 
45  14657  0.00017745 
46  12410  0.00015024 
47  10299  0.00012469 
48  8666  0.00010492 
49  7355  0.00008904 
50  5901  0.00007144 
51  5017  0.00006074 
52  4227  0.00005117 
53  3452  0.00004179 
54  2888  0.00003496 
55  2470  0.0000299 
56  2012  0.00002436 
57  1626  0.00001969 
58  1391  0.00001684 
59  1135  0.00001374 
60  924  0.00001119 
61  840  0.00001017 
62  694  0.0000084 
63  534  0.00000646 
64  498  0.00000603 
65  372  0.0000045 
66  316  0.00000383 
67  286  0.00000346 
68  224  0.00000271 
69  197  0.00000238 
70  160  0.00000194 
71  125  0.00000151 
72  86  0.00000104 
73  79  0.00000096 
74  94  0.00000114 
75  70  0.00000085 
76  64  0.00000077 
77  38  0.00000046 
78  42  0.00000051 
79  27  0.00000033 
80  33  0.0000004 
81  16  0.00000019 
82  18  0.00000022 
83  19  0.00000023 
84  14  0.00000017 
85  6  0.00000007 
86  4  0.00000005 
87  9  0.00000011 
88  4  0.00000005 
89  5  0.00000006 
90  5  0.00000006 
91  1  0.00000001 
92  6  0.00000007 
93  1  0.00000001 
94  3  0.00000004 
95  1  0.00000001 
96  1  0.00000001 
97  2  0.00000002 
102  1  0.00000001 
Total  82600000  1 