# Yahtzee game, probability of getting full house,4 of a kind

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

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 $6\cdot5$ 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

$$6\cdot5\cdot\binom52/6^5=30\cdot10/6^5=25/648\approx0.03858\;,$$

and for four of a kind

$$6\cdot5\cdot\binom51/6^5=30\cdot5/6^5=25/1296\approx0.01929\;.$$

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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