The sum is even, so are both dice even?

Question

You roll two fair six-sided dice. Someone tells you truthfully that the sum of the two dice is even. Given that, what is the probability that both dice show even numbers?

Accepted Answer

A sum is even in exactly two ways: both dice even, or both dice odd. Each die is even with probability 1/2 and odd with probability 1/2, independently. - Both even: 3 	imes 3 = 9 outcomes (each die from \{2,4,6\}). - Both odd: 3 	imes 3 = 9 outcomes (each die from \{1,3,5\}). So there are 9 + 9 = 18 outcomes with an even sum, and 9 of them have both dice even: P(	ext{both even} \mid 	ext{sum even}) = \frac{9}{18} = \frac{1}{2}. The even-sum clue splits evenly between "both even" and "both odd," so it is a clean coin flip. Takeaway. Translate a clue (even sum) into the outcomes that satisf…

The sum is even, so are both dice even?

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