Many dice games like Yahtzee or Yamslam are based on making the best poker hand out of a roll of 5 dice. Often there is an opportunity to improve the hand by selectively re-rolling the dice, and sometimes there are constraints added to what hands can be scored. But in any case, a good starting point is to calculate the relative probabilities of getting one of the basic poker hands on one roll.
Ranking poker dice hands by odds mostly follow the same order as poker hands dealt from a standard 52 card deck (except for 5 of a kind which doesn’t exist in a standard poker hand). Some differences are inevitable since the number of possible 5 card poker hands is
(52,5) = 52! / (5! * 47!) = 2,598,960
and the number of possible 5-die rolls is only
6 ^ 5 = 7776
In the following, to find the odds of plain old poker dice hands, I will simply count the number of ways each can happen and then divide by 7776.
The pattern here is AAAAA.
There are only 6 choices for A, so the probability is 6/7776 = 0.08%
The pattern here is AAAAB.
There are 6 choices for A, and once A is chosen there are 5 choices left for B. But since A and B are distinguishable, we count all combinations, and B can be in (5,1) = 5 possible places, so the probability is
6 * 5 * 5 / 7776 = 150 / 7776 = 1.93%
In a departure from standard poker hands, the next hand in poker dice is a straight. The pattern here is ABCDE where all die are distinct numbers. There’s a trick – in poker dice there are only two straights: 12345 and 23456. Since all dice are distinguishable, there are (5 1) places the first die can go, (4 1) for the second, (3 1) for the third, (2 1) for the fourth, and the last is determined. The probability is then 2 * (5 * 4 * 3 * 2) /7776 = 240/7776 = 3.09%
The pattern for a full house is AAABB with 6 choices for A and 5 remaining choices for B. The A’s can be arranged in (5 3) different ways and then the B’s are determined. (Or you can say there are (5 2) different ways to arrange the B’s and the A’s are determined; you get the same answer.) The probability is 6 * 5 * (5 3) / 7776 = 300 / 7776 = 3.86%
The pattern for three of a kind is AAABC, and you might think there are 6 choices for A, 5 choices for B, and 4 choices for C. But since B and C are indistinguishable, there are 6 choices for A and (5 2) choices for assignments to B and C. There are (5 3) ways to arrange the A’s, and (2 1) ways to arrange B in the remaining slots and then the position of C is determined. This gives a probability of 6 * (5 2) * (5 3) * (2 1) / 7776 = 1200 / 7776 = 15.43%.
The pattern for two pair is AABBC, and you might think there are 6 choices for A, 5 choices for B, and 4 choices for C. But since the pairs are themselves indistinguishable, it’s not really 6 * 5, it’s actually (6 2) possibilities for assignments to A and B, with 4 choices for C. There are also (5 2) ways to arrange the A’s, and (3 2) ways to arrange the B’s in the remaining spots, and then the position of C is determined. The probability is then (6 2) * 4 * (5 3) * (3 2) / 7776 = 1800 / 7776 = 23.15%.
The pattern for one pair is AABCD. This one has 6 choices for A, and (5 3) ways to assign values to B, C, and D. There are (5 2) ways to arrange the pair in 5 slots, (3 1) places to put B, (2 1) places to put C, and then the position of D is determined. The probability is then 6 * (5 3) * (5 2) * (3 1) * (2 1) / 7776 = 3600 / 7776 = 46.30%
The unofficial word for a nothing roll is Bupkiss. Interestingly, the chances of a nothing roll is pretty small! In fact, there are really only four nothing rolls: 12346, 12356, 12456, and 13456. Each one can be arranged in (5 1) (4 1) (3 1) (2 1) ways for a probability of 4 * 5 * 4 * 3 * 2 / 7776 = 480 / 7776 = 6.17%. So in poker dice, Bupkiss should actually beat Three of a Kind.
In conclusion, the table of odds of the standard poker hands in poker dice are given by
---------------------------------------------- Hand Count Probability ---------------------------------------------- Five of a Kind 6 0.08% Four of a Kind 150 1.93% Straight 240 3.09% Full House 300 3.86% Bupkiss 480 6.17% Three of a Kind 1200 15.43% Two Pair 1800 23.15% One Pair 3600 46.30% ---------------------------------------------- Total 7776 100.00% ----------------------------------------------