Expected first die given the two dice sum to nine

Question

You roll two fair six-sided dice. A friend tells you their sum is 9, but nothing else. Given that the sum is 9, what is the expected value of the first die?

Accepted Answer

This is a conditional expectation: you want E[	ext{first die} \mid 	ext{sum} = 9]. There are two clean ways to see it. Method 1, list the possibilities. The ordered pairs that sum to 9 are (3,6), (4,5), (5,4), (6,3), all equally likely. The first die takes the values 3, 4, 5, 6, each once, so E[	ext{first die} \mid 	ext{sum} = 9] = \frac{3 + 4 + 5 + 6}{4} = \frac{18}{4} = 4.5. Method 2, symmetry (the slick way). The two dice play identical roles, so given any fixed sum they have the same conditional expectation: E[	ext{first} \mid 	ext{sum}] = E[	ext{second} \mid 	ext{sum}]. But the t…

Expected first die given the two dice sum to nine

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