Expected glasses-wearers in a group pick

Question

A class has 30 students, of whom 10 wear glasses. Six students are chosen at random for a team. On average, how many of the six wear glasses?

Accepted Answer

Use linearity of expectation: give each glasses-wearer a chance of being chosen, then add. Take any one of the 10 glasses-wearers. A random team of 6 out of 30 includes any given student with probability \frac{6}{30}=\frac{1}{5}. Each such student contributes 	frac15 on average. Add over all 10 of them: 10	imes\frac{1}{5}=2. Same as sample size times the target fraction: 6	imes\frac{10}{30}=2. Takeaway. Expected count equals draws times target fraction. A third of the class wears glasses and you drew six, so two wear glasses on average.

Expected glasses-wearers in a group pick

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