Question

In a game of yahtzee, where 5 dice are tossed simultaneously, find the probability of getting (a) 2 pairs: (b) 4 of a kind.

Answer 1

(a) The probability of two pairs is approximately 5/6.

(b) Probability of getting at least 4 of a kind is 2%

(a) What is the probability of getting 2 pairs?

If the dice are fair (and we will assume that all of them are), then each of these outcomes is equally likely. Since there are six possible outcomes, the probability of obtaining any side of the dice is 1/6. The probability of rolling a 5 is 5/6, the probability of rolling a 2 pairs is 5/6, and so on.

(b) What is the probability of getting 4 of a kind?

Total number of possible outcomes = $6^5$ (Each of the 5 dice has 6 possibilities) = 7776

The number of possible outcomes which are four of a kind but not five of a kind:

5 (5 options for which 4 dice are included) * 6 (6 options for the digit of four dice) * 5 (five options for the digit of the fifth dice, can’t be the same as the other 4) = 150

Number of outcomes which are 5 of a kind = 6 (could be all 1s, all 2s, all 3s, all 4s, all 5s, all 6s)

So

Probability of getting exactly 4 of a kind (not 5 of a kind) = 150/7776 = 25/1296 = approx 1.93%

Probability of getting at least 4 of a kind

= 156/7776

= 13/648

≈ 2%

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