Two dice, told at least one is a six

Question

You roll two fair six-sided dice, a red one and a blue one, out of your sight.

Given that at least one die shows a six, what is the probability that both dice show sixes? How does the answer change if you are instead told the red die shows a six?

Accepted Answer

The sample space is the 36 equally likely ordered pairs (red, blue). Case 1: "at least one six." Count the outcomes with a six on at least one die. There are 6 with the red die a six and 6 with the blue die a six, but the pair (6,6) is counted in both, so by inclusion-exclusion there are 6 + 6 - 1 = 11 such outcomes. Exactly one of them, (6,6), is a double six: P(	ext{both six} \mid 	ext{at least one six}) = \frac{1}{11}. Case 2: "the red die is a six." Fixing the red die at six leaves 6 equally likely outcomes for the blue die. One of them, blue = 6, gives a double six: P(	ext{both six} \…

Two dice, told at least one is a six

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