Probability of rolling doubles

Question

You roll two fair six-sided dice. What is the probability of rolling doubles (both dice showing the same face)? Is rolling a specific double, say two fours, as likely as rolling a specific mix, say a three and a five?

Accepted Answer

On the 36-cell grid of ordered outcomes, the doubles are the diagonal: (1,1),\ (2,2),\ (3,3),\ (4,4),\ (5,5),\ (6,6), which is 6 outcomes. So P(	ext{doubles}) = \frac{6}{36} = \frac{1}{6}. The specific-pair subtlety. A specific double like two fours is a single ordered cell (4,4), probability 1/36. But "a three and a five" can happen two ways, (3,5) and (5,3), probability 2/36 = 1/18. So an unordered mixed pair is twice as likely as any particular double, precisely because the two dice can be arranged in two orders when they differ but only one when they match. This is why, of the 36 ordered…

Probability of rolling doubles

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