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In the game of Yahtzee, five dice are tossed simultaneously. Find the probability of getting

a. full house

b. 4 of a kind

Bases on wikipedia

Full House = A three-of-a-kind and a pair

Four-Of-A-Kind = At least four dice showing the same face

And total number of cases is 6^5=7776

I really don't understand this game,

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dice pattern probability calculator –  MJD May 15 '13 at 15:43
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2 Answers 2

up vote 2 down vote accepted

The answer depends on whether you want to count only the throws that satisfy only the requirements for a full-house/four-of-a-kind entry, or also the throws with all five dice the same, which you could use either in the full-house/four-of-a-kind row or in the Yahtzee (five-of-a-kind) row. The difference is the probability of a Yahtzee, which is $1$ in $6^4=1296$; I'll calculate the probabilities excluding five of a kind.

In both cases, full house and four of a kind, there are $\binom62$ different choices for the numbers. For full house, there are $\binom52$ choices for the positions and for four of a kind there are $\binom51$. Thus the probability for a full house (excluding five of a kind) is

$$\binom62\binom52/6^5=15\cdot10/6^5=25/1296\approx0.01929\;,$$

and for four of a kind

$$\binom62\binom51/6^5=15\cdot5/6^5=25/2592\approx0.009645\;.$$

Note that these are just the probabilities for the first throw, which I think is what you asked for; the more interesting part of the game is deciding which dice to keep to optimize the chances on subsequent throws.

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Actually, both calculations are off by a factor of 2. The reason is that (6¦2) does not regard order. Hence, out of a set of 6 numbers, S={1,2,3,4,5,6}, six choose 2, that is to say, (6¦2)=6!/2!(6-2)!=15, will include the pair of numbers (1,2) for example, but not (2,1). However, (1,2) represents rolling, for example, four ones and one two but does not represent at the same time rolling four two’s and a single one. Hence, there are 2×(6¦2) different choices for numbers. Similarly for the full house calculation, the equation should be multiplied by 2.

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