Expected women on a random committee

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A group has 8 women and 7 men, 15 people in total. A committee of 5 is chosen at random. On average, how many women end up on the committee?

Accepted Answer

The clean way is to think about each seat, or better, each woman, separately, then add up. This is the "linearity of expectation" idea: the average of a total is the sum of the averages, even when the pieces are linked. Focus on any one of the 8 women. By symmetry, a randomly chosen committee of 5 out of 15 gives each person the same chance of being included, namely \frac{5}{15}=\frac{1}{3}. That is her chance of being on the committee. Treat "she is on it" as counting 1, and "she is not" as counting 0. Her average contribution to the count of women is therefore 	frac13. Now add this across …

Expected women on a random committee

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