Expected first draw given two draws sum to twelve

Question

You draw two numbers, each chosen uniformly and independently from $\{1, 2, \dots, 10\}$ (with replacement). You are told the two draws add up to $12$ .

Given that the sum is $12$ , what is the expected value of the first draw?

Accepted Answer

You want E[	ext{first draw} \mid 	ext{sum} = 12]. The two draws come from the same distribution and are interchangeable, so symmetry does the work. Because the two draws play identical roles, they share the same conditional expectation given the sum: E[	ext{first} \mid 	ext{sum} = 12] = E[	ext{second} \mid 	ext{sum} = 12]. And the two conditional expectations must add to the sum: E[	ext{first} \mid 	ext{sum} = 12] + E[	ext{second} \mid 	ext{sum} = 12] = 12. Two equal numbers adding to 12 are each 6, so E[	ext{first draw} \mid 	ext{sum} = 12] = 6. You can confirm by listing: the or…

Expected first draw given two draws sum to twelve

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