Chance the first coin is heads given seven of ten

Question

You flip ten fair coins and are told that exactly 7 of them came up heads. Given that there are 7 heads in total, what is the probability that the first coin is heads?

Accepted Answer

Write Xi = 1 if coin i is heads and 0 if tails. The probability that coin 1 is heads, given the total, equals the conditional expectation of its indicator: P(	ext{coin 1 heads} \mid 	ext{total} = 7) = E[X1 \mid X1 + \cdots + X{10} = 7]. Symmetry pins it down. All ten coins are interchangeable, so each has the same conditional expectation given the total: E[X1 \mid 	ext{total} = 7] = E[X2 \mid 	ext{total} = 7] = \cdots = E[X{10} \mid 	ext{total} = 7]. These ten equal numbers must add to the total number of heads: E[X1 \mid 	ext{total} = 7] + \cdots + E[X{10} \mid 	ext{total} = 7] = 7. T…

Chance the first coin is heads given seven of ten

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