The chance of rolling doubles with two dice

If you roll two normal six-sided dice there's a 1 in 6 chance of rolling doubles. But what if you roll two dice that don't have the same number of sides? My sons are big fans of the game Dungeons and Dragons, so they have a fairly big collection of dice, most of which don't have 6 sides. They have dice with 3, 4, 5, 6, 7, 8, 10, 12,14, 16, 20, 24, 30, 34, 50 and 100 sides. Maybe others, also. That's all that they could think of when I recently asked them. In any event, there are definitely lots of ways to pick a pair of dice to roll that don't necessarily have the same number of sides each.

Suppose that you have two dice that have a sides and b sides each with ab. If we roll these two dice then the probability of rolling the same number of both dice is

a / (a x b) = 1 / b

Note that a, the size of the smaller die doesn't appear there, so that this probability is determined just by the size of the larger die. If you roll a four-sided die and a twenty-sided die, there's a 1 in 20 chance of rolling doubles. If you change that to a twelve-sided die and a twenty-sided die, there's still a 1 in 20 chance of rolling doubles. I found that to be a somewhat surprising result.

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