One die's expected value given three dice total ten

Question

You roll three fair six-sided dice and are told their total is 10. Given that the total is 10, what is the expected value of any single one of the three dice?

Accepted Answer

You want E[	ext{one die} \mid 	ext{total} = 10]. The direct route would be to list every triple summing to 10, but symmetry gives the answer instantly. The three dice play identical roles, so given any fixed total they share the same conditional expectation: E[	ext{die}1 \mid 	ext{total}] = E[	ext{die}2 \mid 	ext{total}] = E[	ext{die}3 \mid 	ext{total}]. These three equal conditional expectations must add up to the total, because their sum is the total's conditional expectation given itself: E[	ext{die}1 \mid 	ext{total} = 10] + E[	ext{die}2 \mid 	ext{total} = 10] + E[	ext{die}3 …

One die's expected value given three dice total ten

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