Some modest Yahtzee goals

Probability Level 3

If the probability of getting at least one pair of matching dice when rolling five 6-sided dice is $\frac{a}{b}$ , where $a$ and $b$ are positive co-prime integers, then what is $a+b$ ?

Note: Three-of-a-kind, four-of-a-kind, full house, yahtzee, or two pairs would all count as "at least one pair of matching dice." However, a straight ( $\{1,2,3,4,5\}$ or $\{2,3,4,5,6\}$ ) or a small straight would not count.

Check out The anti-Yahtzee first to make finding the solution to this problem trivial.

The answer is 103.

1 solution

Andy Hayes
Dec 6, 2015

The solution to the anti-Yahtzee problem, in which the goal was to get five distinct dice rolls, is $\frac{6}{6}\times\frac{5}{6}\times\frac{4}{6}\times\frac{3}{6}\times\frac{2}{6}=\frac{5}{54}$ . Alternatively, $\Large\frac{\binom{6}{5}\times 5!}{6^5}=\normalsize\frac{5}{54}$ .

This problem asks for the complement event of the anti-Yahtzee problem. Therefore, the probability is $1-\frac{5}{54}=\frac{49}{54}$ , and so $a+b=\boxed{103}$ .

Some modest Yahtzee goals

The answer is 103.

1 solution

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