Probability  Dice
These bets can be made in both sic bo and chuck a luck.
[Bluejay adds: Uh, yeah, but I think the point was that it was for charity. What’s more fun: Donating £1.00 to charity and getting nothing back but the good feeling of helping out, or donating £1.00 and getting the good feeling plus the longshot chance of winning a car?]
 Five of a kind: 6/6^{5} = 0.08% (obvious)
 Four of a kind: 5*6*5 = 1.93% (five possible positions for the singleton * 6 ranks for the four of a kind * 5 ranks for the singleton).
 Full house: combin(5,3)*6*5/6^{5} = 3.86% (combin(5,3) positions for the three of a kind * 6 ranks for the three of a kind * 2 ranks for the pair).
 Three of a kind: COMBIN(5,3)*COMBIN(2,1)*6*COMBIN(5,2) / 6^{5} = 15.43%. (combin(5,3) positions for the three of a kind * combin(2,1) positions for the larger of the singletons * 6 ranks of the three of a kind * combin(5,2) ranks for the two singletons.
 Two pair: COMBIN(5,2)*COMBIN(3,2)*COMBIN(6,2)*4 / 6^{5} = 23.15% (combin(5,2) positions for the higher pair * combin(3,2) positions for the lower pair * combin(6,4) ranks for the two pair * 4 ranks for the singleton.
 Pair: COMBIN(5,2)*fact(3)*6*combin(5,3) / 6^{5} = 46.30% (combin(5,2) positions for the pair * fact(3) positions for the three singletons * 6 ranks for the pair * combin(5,3) ranks for the singletons.
 Straight: 2*fact(5) / 6^{5} = 3.09% (2 spans for the straight {15 or 26} * fact(5) ways to arrange the order).
 Nothing: ((COMBIN(6,5)2)*FACT(5)) / 6^{5} = 6.17% (combin(6,5) ways to choose 5 ranks out of six, less 2 for the straights, * fact(5) ways to arrange the order.
w=1(1p)^{n}
1w = (1p)^{n}
log(1w) = log((1p)^{n})
log(1w) = n*log(1p)
n= log(1w)/log(1p)
So in your example n = log(1.5) / log(1(1/36)) = log(0.5) / log(35/36) = 24.6051. So if the probability of success is 50% in 24.6 rolls it must be slightly less in 24 rolls.
3 dice: 25.93%
4 dice: 48.77%
5 dice: 66.13%.
Let B = Rolling a 6 with randomly chosen die
Answer = Pr(A given B) = Pr(A and B)/pr(B) = ((2/3)*(1/6))/((2/3)*(1/6)+(1/3)*1) = (2/18)/((2/18)+(6/18)) = 1/4.
Ex(x) = 1 + (5/6)*ex(x) + (1/6)*ex(y), and
Ex(y) = 1 + (5/6)*ex(x)
Solving for these two equations...
Ex(x) = 1 + (5/6)*ex(x) + (1/6)*( 1 + (5/6)*Ex(x))
Ex(x) = 7/6 + (35/36)*Ex(x)
(1/36)*Ex(x) = 7/6
Ex(x) = 36*(7/6) = 42
So the average wait time for two consecutive twos is 42 rolls.
I have the same type of problem, only the expected flips to get two heads, in my site of math problems, see problem 128.
Probability of a Pair or More
Rolls  Probability 
2 rolls  16.67% 
3 rolls  44.44% 
4 rolls  72.22% 
5 rolls  90.74% 
6 rolls  98.46% 
(8/36)^{n} = 1/41,400,000
log((8/36)^{n}) = log(1/41,400,000)
n × log(8/36) = log(1/41,400,000)
n = log(1/41,400,000)/log(8/36)
n = 7.617 / 0.65321
n = 11.6608
So there you go, the probability of hitting the SuperLotto is the same as rolling a seven or eleven 11.66 times in a row. For those who can’t comprehend a partial throw I would rephrase as the probability falls between 11 and 12 consecutive rolls.
If two dice are rolled over and over, until either of the following events happen, then which is more likely to happen first:
 A total of six and eight is rolled, in either order, with duplicates allowed.
 A total of seven is rolled twice.
Somebody offered me a bet that that the six and eight would occur first. I accepted because seven is the most likely total. However, I lost $2,500 doing this over and over. What are the odds?
Two Sevens before Six and Eight Bet
Relavant Rolls  Probability  Formula  Outcome 
6,8  0.142045  (5/16)*(5/11)  Lose 
8,6  0.142045  (5/16)*(5/11)  Lose 
6,7,8  0.077479  (5/16)*(6/11)*(5/11)  Lose 
7,6,8  0.053267  (6/16)*(5/16)*(5/11)  Lose 
8,7,6  0.077479  (5/16)*(6/11)*(5/11)  Lose 
7,8,6  0.053267  (6/16)*(5/16)*(5/11)  Lose 
7,7  0.140625  (6/16)*(6/16)  Win 
6,7,7  0.092975  (5/16)*(6/11)*(6/11)  Win 
8,7,7  0.092975  (5/16)*(6/11)*(6/11)  Win 
7,6,7  0.06392  (6/16)*(5/16)*(6/11)  Win 
7,8,7  0.06392  (6/16)*(5/16)*(6/11)  Win 
Basically, the reason the 6 and 8 is the better side is you can hit those numbers in either order: 6 then 8, or 8 then 6. With two sevens there is only one order, a 7 and then another 7.
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 !!!
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%.
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 
However, for practical purposes, there is some stopping point. This is because the happiness money brings is not proportional to the amount. While it is commonly accepted that more money brings more happiness, the richer you get, the less happiness each additional dollar brings you.
I believe a good way to answer this question is to apply the Kelly Criterion to the problem. According to Kelly, the player should make every decision with the goal of maximizing the expected log of his bankroll after the wager. To cut to the end of this (I cut out a lot of math), the player should keep doubling until the wager amount exceeds 96.5948% of his total wealth. Wealth should be defined as the sum of the amount won plus whatever money the player had before he made the first wager. For example, if the player had $100,000 to start with, he should keep doubling up to 23 times, to a win of $4,194,304. At that point the player’s total wealth will be $4,294,304. He will be asked to wager 4,194,304/4,294,304 = 96.67% of his total wealth, which is greater than the 96.5948% stopping point, so he should quit.
p = Prob(6 on first roll) + Prob(no 6 on first roll)*Prob(no 7 on second roll)*p.
This is because, if neither player wins after the first two rolls, the game is back to the original state, and the probability of player A winning remains the same.
So, we have:
p = (5/36) + (31/36)×(30/36)×p
p = 5/36 + (930/1296)×p
p * (1(930/1296)) = 5/36.
p * (366/1296) = 5/36
p = (5/36)×(1296/366) = 30/61.
NonDistinct Dice Combinations
Dice  Combinations 
1  6 
2  21 
3  56 
4  126 
5  252 
6  462 
7  792 
8  1287 
9  2002 
10  3003 
11  4368 
12  6188 
13  8568 
14  11628 
15  15504 
16  20349 
17  26334 
18  33649 
19  42504 
20  53130 
21  65780 
22  80730 
23  98280 
Credit to Alan Tucker, author of Applied Combinatorics.
Two sevens in a row?
Three sevens in a row?
Four sevens in a row?
Thanks for your time :).
It is a little easier getting a specified sequence of sevens starting with the first roll, or ending with the last, because the sequence is bounded on one side. Specifically, the probability of getting a sequence of s sevens, starting with the first roll, or ending with the last, is (1/6)^{s} × (5/6). The 5/6 term is because you have to get a non7 at the open end of the sequence.
The probability of starting a sequence of s sevens at any point in the middle of the sequence is (1/6)^{s} × (5/6)^{2}. We square the 5/6 term, because the player must get a non7 on both ends of the sequence.
If there are r rolls, there will be 2 places for an inside sequence, and rn1 places for a run of n sevens. Putting these equations in a table, here is the expected number of runs of sevens, from 1 to 10. The "inside" column is 2*(5/6)*(1/6)^{r}, and the "outside" column is (179r)*(5/6)^{2}*(1/6)^{r}, where r is the number of sevens in the run. So, we can expect 3.46 runs of two sevens, 0.57 runs of three sevens, and 0.10 runs of four sevens.
Expected Runs of Sevens in 180 Rolls
Run  Inside  Outside  Total 
1  0.277778  20.601852  20.87963 
2  0.046296  3.414352  3.460648 
3  0.007716  0.565844  0.57356 
4  0.001286  0.093771  0.095057 
5  0.000214  0.015539  0.015754 
6  0.000036  0.002575  0.002611 
7  0.000006  0.000427  0.000433 
8  0.000001  0.000071  0.000072 
9  0  0.000012  0.000012 
10  0  0.000002  0.000002 

If you limit yourself to the regular polygons, and want every face to have the same probability, then you are limited to the platonic solids. However, if you can lift the regular polygon requirement, then you can add the 13 Catalan solids as well.
To answer your other question, no, I have never seen a game actually in a casino that used any dice other than cubes. About ten years ago I saw a game demonstrated at a gaming show in Atlantic City that I think used a Rhombic triacontahedron, one of the Catalan solids, but I don’t think it ever made it to a casino floor. There is a game I see year after year at the Global Gaming Expo that uses a spinning top (like a dreidel), but alas, I’ve never seen that in a casino either.
Combinations in 4d6L
Outcome  Combinations 
3  1 
4  4 
5  10 
6  21 
7  38 
8  62 
9  91 
10  122 
11  148 
12  167 
13  172 
14  160 
15  131 
16  94 
17  54 
18  21 
Total  1296 
The mean result is 12.2446, and the standard deviation is 2.8468.
If you rephrase the question to be what is the probability of rolling five 6’s before a 12, then the answer is (6/7)^{5} = 46.27%. With four rolls it is (6/7)^{4} = 53.98%. So there is no number of 7’s before a 12 that is exactly 50/50. If you’re looking for a good sucker bet, suggest you can either roll four 7’s before a 12, or a 12 before five 7’s.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
OneDie Probabilities
Dice Total  1  2  3  4  5  6  
One Die  0  0  0  0  0  1  1  1  1  1  1  0  0  0  0  0 
This represents the number of combinations for rolling a 1 to 6 with one die. I know, pretty obvious. However, stick with me. For two dice, add another row to the bottom, and for each cell take the sum of the row above and the five cells to the left of it. Then add another five dummy zeros to the right, if you wish to keep going. This represents the combinations of rolling a total of 2 to 12.
Two Dice Probabilities
Dice Total  2  3  4  5  6  7  8  9  10  11  12  
One Die  0  0  0  0  0  1  1  1  1  1  1  0  0  0  0  0  0  0  0  0  0 
Two Dice  0  0  0  0  0  1  2  3  4  5  6  5  4  3  2  1  0  0  0  0  0 
For three dice, just repeat. This will represent the number of combinations of 3 to 18.
To get the probability of any given total, divide the number of combinations of that total by the total number of combinations. In the case of three dice, the sum is 216, which also easily found as 6^{3}. For example, the probability of rolling a total of 13 with three dice is 21/216 = 9.72%.
So for d dice, you’ll need to work your way up through 1 to d1 dice. This is very easily accomplished in any spreadsheet.
(35/36)^{r} = 0.5
log(35/36)^{r} = log(0.5)
r × log(35/36) = log(0.5)
r = log(0.5)/log(35/36)
r = 24.6051
So there isn’t a round answer. The probability of rolling a 12 in 24 rolls is 1(35/36)^{24} = 49.14%. The probability of rolling a 12 in 25 rolls is 1(35/36)^{25} = 50.55%.
If you want to make a bet on this, say you can roll a 12 in 25 rolls, or somebody else can’t in 24 rolls. Either way you’ll have an advantage at even money.
For those unfamiliar with the game, both the attacker and defender will roll 1 to 8 dice, according to how many armies they each have at that point in a battle. The higher total shall win. A tie goes to the defender. If the attacker loses, he will still retain one army in the territory where he initiated the attack. For this reason, he must have at least two armies to attack, so if he wins one can inhabit the conquered territory and one can stay behind.
The following table shows the probability of an attacker victory according to all 64 combinations of total dice.
Probability of Attacker Win
Attacker  Defender  

1 Army  2 Armies  3 Armies  4 Armies  5 Armies  6 Armies  7 Armies  8 Armies  
2  0.837963  0.443673  0.152006  0.035880  0.006105  0.000766  0.000071  0.000005 
3  0.972994  0.778549  0.453575  0.191701  0.060713  0.014879  0.002890  0.000452 
4  0.997299  0.939236  0.742831  0.459528  0.220442  0.083423  0.025450  0.006379 
5  0.999850  0.987940  0.909347  0.718078  0.463654  0.242449  0.103626  0.036742 
6  0.999996  0.998217  0.975300  0.883953  0.699616  0.466731  0.259984  0.121507 
7  1.000000  0.999801  0.994663  0.961536  0.862377  0.685165  0.469139  0.274376 
8  1.000000  0.999983  0.999069  0.989534  0.947731  0.843874  0.673456  0.471091 
The next table shows the expected gain by the attacker, defined as pr(attacker wins)*(defender dice)+pr(defender wins)*(attacker dice 1). It shows the greatest expected gain is to attack with 8 against an opponent with 5.
Net Gain of Attacker Win
Attacker  Defender  

1 Army  2 Armies  3 Armies  4 Armies  5 Armies  6 Armies  7 Armies  8 Armies  
2  0.675926  0.331019  0.391976  0.820600  0.963370  0.994638  0.999432  0.999955 
3  0.918982  1.114196  0.267875  0.849794  1.575009  1.880968  1.973990  1.995480 
4  0.989196  1.696180  1.456986  0.216696  1.236464  2.249193  2.745500  2.929831 
5  0.999250  1.927640  2.365429  1.744624  0.172886  1.575510  2.860114  3.559096 
6  0.999976  1.987519  2.802400  2.955577  1.996160  0.134041  1.880192  3.420409 
7  1.000000  1.998408  2.951967  3.615360  3.486147  2.221980  0.098807  2.158736 
8  1.000000  1.999847  2.990690  3.884874  4.372772  3.970362  2.428384  0.066365 
The player may reroll previously held dice, if he wishes. For example, if the player's first roll is 33456 and he holds the threes and then has 33555 after the second roll he may keep the fives and reroll the threes on his third roll.
The following table shows the maximum number of dice of the same face for 1 to 20 rolls. The table shows the probability of getting a Yahtzee within three rolls is about 4.6%.
Yahtzee Probabilities
Rolls  Maximum Dice of Same Face  

One  Two  Three  Four  Five  
1  0.092593  0.694444  0.192901  0.019290  0.000772 
2  0.008573  0.450103  0.409022  0.119670  0.012631 
3  0.000794  0.256011  0.452402  0.244765  0.046029 
4  0.000074  0.142780  0.409140  0.347432  0.100575 
5  0.000007  0.079373  0.337020  0.413093  0.170507 
6  0.000001  0.044101  0.263441  0.443373  0.249085 
7  0.000000  0.024501  0.199279  0.445718  0.330502 
8  0.000000  0.013612  0.147462  0.428488  0.410438 
9  0.000000  0.007562  0.107446  0.398981  0.486011 
10  0.000000  0.004201  0.077416  0.362855  0.555528 
11  0.000000  0.002334  0.055317  0.324175  0.618174 
12  0.000000  0.001297  0.039279  0.285674  0.673750 
13  0.000000  0.000720  0.027757  0.249063  0.722460 
14  0.000000  0.000400  0.019543  0.215313  0.764744 
15  0.000000  0.000222  0.013720  0.184883  0.801175 
16  0.000000  0.000124  0.009610  0.157896  0.832371 
17  0.000000  0.000069  0.006719  0.134258  0.858954 
18  0.000000  0.000038  0.004692  0.113753  0.881517 
19  0.000000  0.000021  0.003272  0.096100  0.900607 
20  0.000000  0.000012  0.002280  0.080994  0.916714 
This question is raised and discussed in my forum at Wizard of Vegas.
A bit offtopic, but I've always thought an odd/even set of bets would be a good way to replace the dreaded big 6/8 bets in craps. To give the house an advantage, here are my proposed pay tables and analysis.
Odd Bet
Event  Pays  Combinations  Probability  Return 

3 or 11  1.5  4  0.111111  0.166667 
5 or 9  1  8  0.222222  0.222222 
7  0.5  6  0.166667  0.083333 
Even  1  18  0.500000  0.500000 
Total  36  1.000000  0.027778 
Even Bet
Event  Pays  Combinations  Probability  Return 

2 or 12  3  2  0.055556  0.166667 
4 or 10  1  6  0.166667  0.166667 
6 or 8  0.5  10  0.277778  0.138889 
Odd  1  18  0.500000  0.500000 
Total  36  1.000000  0.027778 
Please note that I claim all rights with this publication.
This question is raised and discussed in my forum at Wizard of Vegas.
 2 or 12: 1000
 3 or 11: 600
 4 or 10: 400
 5 or 9: 300
 6 or 8: 200
He keeps rolling until he gets a total of seven, which ends the bonus. If he rolls a seven on the first roll, then he gets a consolation prize of 700 coins. What is the average coins won per bonus?
Next, here is the probability of each total, assuming no seven:
 2 or 12: 1/30
 3 or 11: 2/30
 4 or 10: 3/30
 5 or 9: 4/30
 6 or 8: 5/30
So, the average win per roll, assuming no seven, is 2*[(1/30)*1000 + (2/30)*600 + (3/30)*400 + (4/30)*300 + (5/30)*200] = 373.33.
The value of the consolation prize is (1/6)*700 = 116.67.
Thus, the average bonus win is 116.67 + 5×373.33 = 1983.33.
What would be the answer to the dice problem in Ask the Wizard column #179, if the players took turns rolling the dice and only the player rolling could advance based on the roll?
Here was the original question posted in column #179: If two dice are rolled over and over, until either of the following events happen, then which is more likely to happen first:
 A total of six and eight is rolled, in either order, with duplicates allowed.
 A total of seven is rolled twice.
Your twist is that the same roll can't help both players. Instead, they take turns rolling and only the one rolling can use the roll.
The answer depends on who rolls first. If the player needing a six and eight rolls first, then he has a probability of winning of 57.487294%. If the player needing two sevens goes first, then the probability the player needing the six and eight wins is 52.671614%. I solved it using a simple Markov Chain process.
This question is asked and discussed in my forum at Wizard of Vegas.
Suppose you have 12 sixsided dice. You roll them and may set aside any dice you wish. You then reroll the other dice. What is the probability of getting a 12ofakind in the two rolls?
There are 58 different types of sequences on the initial roll. The way I identify each is the number of the face in majority, then the number of dice of the face second in total, and so on. For example, a roll of of 3,3,3,3,6,6,6,5,5,2 would be signified as 4321. The following table shows the number of combinations of each sequence, the probability of rolling it, the probability of completing a 12 of a kind in the second roll, and the product of the two. For the probability on the second roll, I assume the player holds the dice that have the greatest total on the initial roll. The lower right cell shows an overall probability of 0.0000037953, which equals 1 in 263,486.
12 Dice Question
Sequence  Combinations  Probability Sequence 
Conditional Probability 
Total Probability 

1200000  6  0.0000000028  1.0000000000  0.0000000028 
1110000  360  0.0000001654  0.1666666667  0.0000000276 
1020000  1,980  0.0000009096  0.0277777778  0.0000000253 
1011000  7,920  0.0000036384  0.0277777778  0.0000001011 
930000  6,600  0.0000030320  0.0046296296  0.0000000140 
921000  79,200  0.0000363840  0.0046296296  0.0000001684 
911100  79,200  0.0000363840  0.0046296296  0.0000001684 
840000  14,850  0.0000068220  0.0007716049  0.0000000053 
831000  237,600  0.0001091519  0.0007716049  0.0000000842 
822000  178,200  0.0000818639  0.0007716049  0.0000000632 
821100  1,069,200  0.0004911837  0.0007716049  0.0000003790 
811110  356,400  0.0001637279  0.0007716049  0.0000001263 
750000  23,760  0.0000109152  0.0001286008  0.0000000014 
741000  475,200  0.0002183039  0.0001286008  0.0000000281 
732000  950,400  0.0004366077  0.0001286008  0.0000000561 
731100  2,851,200  0.0013098232  0.0001286008  0.0000001684 
722100  4,276,800  0.0019647348  0.0001286008  0.0000002527 
721110  5,702,400  0.0026196464  0.0001286008  0.0000003369 
711111  570,240  0.0002619646  0.0001286008  0.0000000337 
660000  13,860  0.0000063672  0.0000214335  0.0000000001 
651000  665,280  0.0003056254  0.0000214335  0.0000000066 
642000  1,663,200  0.0007640635  0.0000214335  0.0000000164 
641100  4,989,600  0.0022921906  0.0000214335  0.0000000491 
633000  1,108,800  0.0005093757  0.0000214335  0.0000000109 
632100  19,958,400  0.0091687624  0.0000214335  0.0000001965 
631110  13,305,600  0.0061125083  0.0000214335  0.0000001310 
622200  4,989,600  0.0022921906  0.0000214335  0.0000000491 
622110  29,937,600  0.0137531436  0.0000214335  0.0000002948 
621111  9,979,200  0.0045843812  0.0000214335  0.0000000983 
552000  997,920  0.0004584381  0.0000035722  0.0000000016 
551100  2,993,760  0.0013753144  0.0000035722  0.0000000049 
543000  3,326,400  0.0015281271  0.0000035722  0.0000000055 
542100  29,937,600  0.0137531436  0.0000035722  0.0000000491 
541110  19,958,400  0.0091687624  0.0000035722  0.0000000328 
533100  19,958,400  0.0091687624  0.0000035722  0.0000000328 
532200  29,937,600  0.0137531436  0.0000035722  0.0000000491 
532110  119,750,400  0.0550125743  0.0000035722  0.0000001965 
531111  19,958,400  0.0091687624  0.0000035722  0.0000000328 
52221  59,875,200  0.0275062872  0.0000035722  0.0000000983 
522111  59,875,200  0.0275062872  0.0000035722  0.0000000983 
444000  693,000  0.0003183598  0.0000005954  0.0000000002 
443100  24,948,000  0.0114609530  0.0000005954  0.0000000068 
442200  18,711,000  0.0085957147  0.0000005954  0.0000000051 
442110  74,844,000  0.0343828589  0.0000005954  0.0000000205 
441111  12,474,000  0.0057304765  0.0000005954  0.0000000034 
433200  49,896,000  0.0229219060  0.0000005954  0.0000000136 
433110  99,792,000  0.0458438119  0.0000005954  0.0000000273 
432210  299,376,000  0.1375314358  0.0000005954  0.0000000819 
432111  199,584,000  0.0916876238  0.0000005954  0.0000000546 
422220  37,422,000  0.0171914295  0.0000005954  0.0000000102 
422211  149,688,000  0.0687657179  0.0000005954  0.0000000409 
333300  5,544,000  0.0025468784  0.0000000992  0.0000000003 
333210  133,056,000  0.0611250826  0.0000000992  0.0000000061 
333111  44,352,000  0.0203750275  0.0000000992  0.0000000020 
332220  99,792,000  0.0458438119  0.0000000992  0.0000000045 
332211  299,376,000  0.1375314358  0.0000000992  0.0000000136 
322221  149,688,000  0.0687657179  0.0000000992  0.0000000068 
222222  7,484,400  0.0034382859  0.0000000165  0.0000000001 
Total  2,176,782,336  1.0000000000  0.0000037953 
Click the button below for the answer.
Here is my solution. (PDF)
This question is asked and discussed in my forum at Wizard of Vegas.
Suppose a fair sixsided die is rolled until a 1, 2, 3, or 6 appears. If a 1, 2, or 3 is the first of these gameending numbers to appear, then you win nothing. If a 6 is the first of these gameending numbers to appear, then you win $1 for every roll of the die. What is the average win of this game?
Click the button below for a couple infinite series formulas that you may find helpful.
Hint 1: Sum for i = 0 to ∞ of n^{i} = 1 / (1n)
Hint 2: Sum for i = 0 to ∞ of i × n^{i} = n / (1n)^{2}
Click the button below for the answer.
Click the button below for the solution.
Suppose a fair sixsided die is rolled until a 1, 2, 3, or 6 appears. If a 1, 2, or 3 is the first of these gameending numbers to appear, then you win nothing. If a 6 is the first of these gameending numbers to appear, then you win $1 for every roll of the die. What is the average win of this game?
Hint 1: Sum for i = 0 to ∞ of n^{i} = 1 / (1n)
Hint 2: Sum for i = 0 to ∞ of i × n^{i} = n / (1n)^{2}
The expected win can be expressed as the sum for i = 0 to ∞ of (1 + i) * (1/3)^{i} * (1/6). =
(1/6) * sum for i = 0 to ∞ of (1/3)^{i} + (1/6) * sum for i = 0 to ∞ of (i * (1/3)^{i}).
Let's evaluate these one at a time.
sum for i = 0 to ∞ of (1/3)^{i} =
1 / (1  (1/3)) =
1 / (2/3) =
3/2
Sum for i = 0 to ∞ of (i * (1/3)^{i}) =
(1/3) / (1  (1/3))^{2} =
(1/3) / (4/9) =
(1/3) * (9/4) =
3/4
Putting it all together, the answer is
(1/6) * (3/2) + (1/6)*(3/4) =
(1/4) + (1/8) =
3/8
This question is asked and discussed in my forum at Wizard of Vegas.
While this could be solved with a long and tedious Markov chain, I prefer an integral solution. I explain how to use this method in my pages on the Fire Bet and Bonus Craps.
Imagine that instead of significant events being determined by the roll of the die, one at a time, consider them as an instant in time. Assume the time between events has a memoryless property, with an average time between events of one unit of time. In other words, the time between events follows an exponential distribution with a mean of 1. This will not matter for purposes of adjudicating the bet, because events still happen one at a time.
Per the Poisson distribution, the probability that any given side of the die has been rolled zero times in x units of time is exp(x/6)*(x/6)^{0}/0! = exp(x/6). Poisson also say the probability of any given side being rolled exactly once is exp(x/6)*(x/6)^{1}/1! = exp(x/6) * (x/6). Thus probability any side has been rolled two or more times in x units of time is 1  exp(x/6)*(1 + (x/6)). The probability that all six sides have been rolled at least twice is (1  exp(x/6)*(1 + (x/6)))^{6}. The probability that at least one side has not been rolled at least twice is equal to:
We need to integrate that over all time to find how much time will go by, on average, where the desired goal has not been achieved.
Fortunately, we can use an integral calculator at this point. For the one linked to, put 1 (1  exp(x/6)*(1 + x/6))^6 dx = apx. 24.1338692 in the text box following "Calculate the integral of" and under custom, set the bound of integration from 0 to ∞.
The answer is 390968681 / 16200000 = apx. 24.13386919753086
This question is asked and discussed in my forum at Wizard of Vegas.
You start with a fair 6sided die and roll it six times, recording the results of each roll. You then write these numbers on the six faces of another, unlabeled fair die. For example, if your six rolls were 3, 5, 3, 6, 1 and 2, then your second die wouldn’t have a 4 on it; instead, it would have two 3s.
Next, you roll this second die six times. You take those six numbers and write them on the faces of yet another fair die, and you continue this process of generating a new die from the previous one.
Eventually, you’ll have a die with the same number on all six faces. What is the average number of transitions from one die to another (or total rolls divided by 6) to reach this state?
Let's label the initial die with letters instead of numbers, to avoid confusion. Let's label each possible die state with letters. For example, AAABBC would mean three of one letter, two of another, and one of a third. The initial state would obviously be ABCDEF.
Let E(ABCDEF) be the expected number of rolls from state ABCDEF.
E(ABCDEF) = 1 + [180 × E(AAAAAB) + 450 × E(AAAABB) + 300 × E(AAABBB) + 1800 × E(AAAABC) + 7200 × E(AAABBC) + 1800 × E(AABBCC) + 7200 × E(AAABCD) + 16200 × E(AABBCD) + 10800 × E(AABCDE) + 720 × E(ABCDEF)]/46656Building on the number of combinations of going from one state to another, the following transition matrix shows how many ways there are for going from each initial state (left column) to each new state. This took a few hours to construct properly, by the way.
Transition Matrix A
State Before 
AAAAAA  AAAAAB  AAAABB  AAABBB  AAAABC  AAABBC  AABBCC  AAABCD  AABBCD  AABCDE  ABCDEF 

AAAAAB  15,626  18,780  9,750  2,500               
AAAABB  4,160  13,056  19,200  10,240               
AAABBB  1,458  8,748  21,870  14,580               
AAAABC  4,098  12,348  8,190  2,580  7,920  10,080  1,440         
AAABBC  794  5,172  8,670  5,020  6,480  17,280  3,240         
AABBCC  192  2,304  5,760  3,840  5,760  23,040  5,760         
AAABCD  732  4,464  4,140  1,680  7,920  14,400  2,520  4,320  6,480     
AABBCD  130  1,596  3,150  1,940  5,280  16,800  3,600  4,800  9,360     
AABCDE  68  888  1,380  760  3,960  11,520  2,520  7,200  14,040  4,320   
ABCDEF  6  180  450  300  1,800  7,200  1,800  7,200  16,200  10,800  720 
I won't go into a long lecture on matrix algebra, except to say let's say matrix B is as follows:
Matrix B
State Before 
AAAAAB  AAAABB  AAABBB  AAAABC  AAABBC  AABBCC  AAABCD  AABBCD  AABCDE  ABCDEF 

AAAAAB  27876  9750  2500  0  0  0  0  0  0  46656 
AAAABB  13056  27456  10240  0  0  0  0  0  0  46656 
AAABBB  8748  21870  32076  0  0  0  0  0  0  46656 
AAAABC  12348  8190  2580  38736  10080  1440  0  0  0  46656 
AAABBC  5172  8670  5020  6480  29376  3240  0  0  0  46656 
AABBCC  2304  5760  3840  5760  23040  40896  0  0  0  46656 
AAABCD  4464  4140  1680  7920  14400  2520  42336  6480  0  46656 
AABBCD  1596  3150  1940  5280  16800  3600  4800  37296  0  46656 
AABCDE  888  1380  760  3960  11520  2520  7200  14040  42336  46656 
ABCDEF  180  450  300  1800  7200  1800  7200  16200  10800  46656 
The answer is the determinant of matrix B to that of matrix A:
Determ(A) = 1,461,067,501,120,670,000,000,000,000,000,000,000,000,000,000
Determ(B) = 14,108,055,348,203,100,000,000,000,000,000,000,000,000,000,000
Determ(B) / Determ(A) = apx. 9.65599148388557
The answer could be approximated as expressed 1  (prob(no 1's) + prob(no 2's) + ... + prob(no 6's)) = 1  6*(5/6)^20 = apx. 0.84349568.
However that would doublesubtract the situations where two different sides never got rolled. There are combin(6,2)=15 ways to choose two sides out of six. The probability that any two given sides never get rolled is (4/6)^20. We need to add those to the probability, because they got subtracted twice in the previous step. So, now we're at 1  6*(5/6)^20 + 15*(4/6)^20 = apx. 0.84800661.
This question is asked and discussed in my forum at Wizard of Vegas.
However, if any group of three sides that had never been rolled would have been triplesubtracted in the first step and tripleadded in the second step. We need to subtract them back out as a state where not all six sides were rolled. There are combin(6,3) = 20 ways to choose three sides out of six. The probability that any specific three sides are never rolled is (3/6)^20. So, now we're at 1  6*(5/6)^20 + 15*(4/6)^20  20*(3/6)^20= apx. 0.847987537.
However, if any group of four sides that had never been rolled would have been quadruplesubtracted in the first step, quadrupleadded in the second step, and quadruple subtracted in the third step. We need to add them back in, because each such state was already subtracted out twice. There are combin(6,4) = 15 ways to choose four sides out of six. The probability that any specific four sides are never rolled is (2/6)^20. So, now we're at 1  6*(5/6)^20 + 15*(4/6)^20  20*(3/6)^20 + 15*(2/6)^20 = apx. 0.84798754089.
However, if all 20 rolls were the same numbers, this situation would have been quintuplesubtracted in the first step, quintupleadded in the first step, quintuplesubtracted in the third step, and quintupleadded in the fourth step. We need to subtracted them back out. So, now we're at 1  6*(5/6)^20 + 15*(4/6)^20  20*(3/6)^20 + 15*(2/6)^20  6*(1/6)^20 = apx. 0.84798754089.
So the answer is 16*(5/6)^20+COMBIN(6,4)*(4/6)^20COMBIN(6,3)*(3/6)^20+COMBIN(6,2)*(2/6)^206*(1/6)^20 = apx. 0.84798754089.
You have two cubes. You can number each side of both dice as you wish, as long as each side is an integer and greater or equal to one. You may repeat the same number on the same die and go as high as you wish. Other than creating standard dice, how can you number them so the probability of any given total is the same as standard dice?
Die 1 = 1,2,2,3,3,4.
Die 2 = 1,3,4,5,6,8.
I'm afraid my solution to this one was pretty much trial and error.
One could use a Markov chain to answer this, but I prefer calculus. The key is that the answer is the same if the time between rolls is the exponentially distributed with a mean of one. That said, the answer can be expressed as the integral from 0 to infinity of:
1(1exp(x/36))^2*(1exp(x/18))^2*(1exp(x/12))^2*(1exp(x/9))^2*(1exp(5*x/36))^2*(1exp(x/6))
You may easily solve such integrals with an integral calculator.
You may also solve any such problem with my Expected Trials Calculator.
Drop Dead is a game played with five standard dice. You begin your turn by rolling all five dice. If none of them are a 2 or 5, you total the dice, add the sum to your point score and roll again. If you do roll any 2s or 5s, your score for the roll is zero. All the dice showing a 2 or 5 are declared dead and set aside. You then roll again with the remaining dice. Play continues with you either scoring points or removing dice. Your turn ends when all your dice are eliminated whereupon you are said to have dropped dead. All your scoring rolls are added together for your final score. High score wins.
What is your expected score for this game?
Let's start with the scenario with one die left and move backwards.
Let the variable a be the expected additional points with one die left.
The average roll that isn't a 2 or 5 is (1+3+4+6)/4 = 7/2.
a = (2/3)×(a + 7/2).
a/3 = 7/3.
a = 7.
Next, let's calculate b, the expected points with two dice left.
b = (2/3)^{2}×(b + 2 × (7/2)) + 2×(2/3)×(1/3)×a.
b = 11.2.
Next, let's calculate c, the expected points with three dice left.
c = (2/3)^{3}×(c + 3× (7/2)) + 3×(2/3)^{2}×(1/3)×b + 3×(2/3)×(1/3)^{2}×b.
c = 1302/95 = 13.705263.
Next, let's calculate d, the expected points with four dice left.
d = (2/3)^{4}×(d + 4× (7/2)) + 4×(2/3)^{3}×(1/3)×c + 6×(2/3)^{2}×(1/3)^{2}×b + 4×(2/3)×(1/3)^{3}×a.
d = 3752/247 = 15.190283.
Finally, let's calculate e, the expected points with five dice left.
e = (2/3)^{5}×(e + 5×(7/2)) + 5×(2/3)^{4}×(1/3)×d + 10×(2/3)^{3}×(1/3)^{2}×c + 10×(2/3)^{2}×(1/3)^{3}×b + 5×(2/3)×(1/3)^{4}×a.
e = 16.064662.
This question is asked and discussed in my forum at Wizard of Vegas.
Roll two dice, a red die and a blue die, over and over. Keep track of the sum of rolls for each die. What is the expected number of roles until these two cumulative totals are equal?
It is hard to explain why the answer is infinity. To make matters more confusing and paradoxical, the probability the totals ever being equal is 1.
The following table shows the probability the totals will be the same for the first time after 1 to 16 rolls.
Probability Equal Totals for First Time
Rolls  Probability 

1  0.166667 
2  0.112654 
3  0.092850 
4  0.080944 
5  0.072693 
6  0.066539 
7  0.061722 
8  0.057819 
9  0.054573 
10  0.051819 
11  0.049443 
12  0.047367 
13  0.045532 
14  0.043895 
15  0.042423 
16  0.041089 
Excel shows a very close fit to this curve is y = 0.1784*x1.011, where x = number of rolls and y = probability.
The sum of this infinite series is infinity.
This question is asked and discussed in my forum at Wizard of Vegas.
Five red dice and five blue dice are rolled. What is the probability the roll is the same for both dice, without regard to order. For example, both rolls are 12336.
The following the table shows for any type of roll:
 The number of different ways this roll can be achieved. For example, for a full house, there are six combinations for the three of a kind and five left for the pair, for a total of 30 different full houses.
 The number of orders. For example, for a full house, there are combin(5,3)=10 ways to choose three out of five dice for the three of a kind. The other two must have the pair.
 The number of ways the given hand can be rolled. This is the product for the first two columns. For example, there are 30 * 10 = 300 ways to roll a full house.
 The probability of the hand. For example, for a full house the probability is 300/6^{5} = 0.038580.
 The probability both rolls are the same and of the given hand. This is the probability from column four squared divided by the second column. For example, the probability two rolls are both a full house is 0.038580^{2}. However, the probability they are the same house is 1/30. So, the probability both rolls are the same full house is 0.038580^{2}/30 = 0.00004961.
The lower right cell shows the total probability both rolls are the same is 0.00635324.
Matching Roll
Type of Roll 
Different Types 
Orders  Total Combinations 
Probability One Roll 
Probability Two Rolls 


Five of a kind  6  1  6  0.00077160  0.00000010  
Four of a kind  30  5  150  0.01929012  0.00001240  
Full house  30  10  300  0.03858025  0.00004961  
Three of a kind  60  20  1,200  0.15432099  0.00039692  
Two pair  60  30  1,800  0.23148148  0.00089306  
Pair  60  60  3,600  0.46296296  0.00357225  
Five singletons  6  120  720  0.09259259  0.00142890  
Total  7,776  1.00000000  0.00635324 
A sixsided die is rolled until either of the following events happen:
A) Any side has appeared six times.
B) Every side has appeared at least once.
What is the probability event A occurs first?
To answer this one as I did, using calculus, I recommend an integral calculator like the one at integralcalculator.com/.
Here is my solution (PDF).
This problem is asked (in slightly different words) and discussed in my forum at Wizard of Vegas.
You wish to play a game that requires an ordinary sixsided die. Unfortunately, you lost the die. However, you have four index cards, which you may mark any way you like. The player must choose two cards randomly from the four, without replacement, and take the sum of the two cards.
How can you number the cards so that the sum of two different cards represents the roll of a die?
Number them 0, 1, 2, and 4.
There are six ways to draw two out of four cards, as follows.
 0+1 = 1
 0+2 = 2
 1+2 = 3
 0+4 = 4
 1+4 = 5
 2+4 = 6
This question is asked and discussed in my forum at Wizard of Vegas.
A sixsided die is rolled over and over until the sum of rolls is 13 or greater. What is the mean, median and mode of the final total?
Median = 14
Mode = 13
I had to use a Markov Chain for this one. The following table shows the probability of each final total according to the running sum in the left column. Start with the obvious cases for totals of 13 to 18. Then, for running sums of 0 to 12, take the average of the six cells below.
The probabilities for the initial state can be found in the first row for a sum of 0.
Markov Chain
Sum of Rolls  13  14  15  16  17  18 

0  0.279263  0.236996  0.192313  0.145585  0.097371  0.048472 
1  0.290830  0.230791  0.188524  0.143842  0.097114  0.048899 
2  0.293393  0.241931  0.181893  0.139625  0.094943  0.048215 
3  0.289288  0.245178  0.193717  0.133678  0.091410  0.046728 
4  0.280369  0.242560  0.198450  0.146988  0.086950  0.044682 
5  0.268094  0.235687  0.197878  0.153768  0.102306  0.042267 
6  0.253604  0.225827  0.193419  0.155611  0.111500  0.060039 
7  0.360232  0.193566  0.165788  0.133380  0.095572  0.051462 
8  0.308771  0.308771  0.142104  0.114326  0.081919  0.044110 
9  0.264660  0.264660  0.264660  0.097994  0.070216  0.037809 
10  0.226852  0.226852  0.226852  0.226852  0.060185  0.032407 
11  0.194444  0.194444  0.194444  0.194444  0.194444  0.027778 
12  0.166667  0.166667  0.166667  0.166667  0.166667  0.166667 
13  1.000000  0.000000  0.000000  0.000000  0.000000  0.000000 
14  0.000000  1.000000  0.000000  0.000000  0.000000  0.000000 
15  0.000000  0.000000  1.000000  0.000000  0.000000  0.000000 
16  0.000000  0.000000  0.000000  1.000000  0.000000  0.000000 
17  0.000000  0.000000  0.000000  0.000000  1.000000  0.000000 
18  0.000000  0.000000  0.000000  0.000000  0.000000  1.000000 
This question is asked and discussed in my forum a Wizard of Vegas.
As you know, the All bet in craps wins if every total from 2 to 12, except 7, is thrown before a 7. How many rolls on average does it take to win this bet, when it does win?
Here is my solution (PDF).
This question is asked and discussed in my forum at Wizard of Vegas.
I see somebody is claiming to have witnesses 18 consecutive yo's (total of 11) in a row at the craps table. How many rolls, on average, would it take to observe that?
Here is my solution (PDF).
A pair of two fair sixsided dice are rolled over and over until one of the following two events occur:
A) A total of 12 is rolled.
B) A total of 7 is rolled two times consecutively.
Which is more likely to happen first?
[spoiler=Solution]
Let:
 p = Probability the 12 is rolled first from the initial state or anytime the previous roll was not a 7.
 q = Probability the 12 is rolled first when the previous roll was a 7.
This is what is known as a Markov Chain problem.
Before we get to that, recall the probability of rolling a total of 7 is 1/6 and that of a 12 is 1/36.
We can define p and q in terms of each other, as follows:
 (1) p = (1/36) + (6/36)q + (29/36)p
 (2) q = (1/36) + (29/36)p
Let's multiply equation (1) by 36:
36p = 1 + 6q + 29p
(3) 7p = 1 + 6q
Let's substitute the value for q in (2) into (3):
7p = 1 + 6*((1/36) + (29/36)p)
7p = 1 + (1/6) + (29/6)p
42p = 6 + 1 + 29p
13p = 7
q = 7/13
So, the probability of rolling the 12 first is 7/13 =~ 53.85%.
The probability of rolling two consecutive 7's first is thus 46.15%.
Thus, it's more likely the total of 12 is rolled first.