At least one six, but did both dice land on six?

Question

You roll two fair six-sided dice, one red and one blue, out of sight. A friend peeks and truthfully tells you that at least one of the dice shows a six.

Given that, what is the probability that both dice show a six?

Accepted Answer

There are 6 	imes 6 = 36 equally likely outcomes. Count the ones with at least one six. - Rolls where the red die is six: 6 outcomes. - Rolls where the blue die is six: 6 outcomes. - The pair (6,6) was counted twice, so subtract 1. That gives 6 + 6 - 1 = 11 outcomes with at least one six. Only one of them, (6,6), has both sixes: P(	ext{both sixes} \mid 	ext{at least one six}) = \frac{1}{11}. The lone (6,6) competes against ten "exactly one six" outcomes, so both-sixes is quite unlikely even after the clue. Takeaway. "At least one" clues keep every mixed outcome in play; counting the condit…

At least one six, but did both dice land on six?

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