1. In the game of Yahtzee, five dice are rolled simultaneously. Here are some in
ID: 3686444 • Letter: 1
Question
1. In the game of Yahtzee, five dice are rolled simultaneously. Here are some interesting outcomes: three-of-a-kind (three dice show the same number, the other two show different numbers) a full house (three-of-a-kind plus two-of-a-kind) a large straight (the five numbers that show up can be consecutively or- dered all five dice show the same number (a Yahtzee!) . a particular combination say two 2's, one 4, two 6's. Now in games involving dice, it is helpful to think of the five dice as being distinct i.e. we name them as dice 1, dice 2, etc. Determine the number of ways of obtaining each of the five outcomes described above.Explanation / Answer
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A)
There are 6 choose 3 = 20 ways of selecting the three distinct values (if the two singletons were equal then we would have a full house). There are again 3!/2 = 3 arrangements of the values, and each of the resulting 60 combinations can be permuted in 5!/(1!1!3!) = 20 ways. This makes a total of 1200 desired outcomes, and a resulting probability of 25/162.
B)
We now only have 6 choose 2 = 15 ways of selecting two distinct values, and there are 2 distinct arrangements of thee values. The resulting 30 combinations can be permuted in 5!/(2!3!) = 10 ways, for a total of 300 desired outcomes, and a resulting probability of 25/648
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