Quizzes returned at random: how many honor students?

Question

A teacher grades 20 quizzes but forgets whose is whose, so she hands all 20 back in a completely random order, one per student. Exactly 5 of the 20 students are honor students.

On average, how many of the 5 honor students receive their own quiz?

Accepted Answer

We only care about the 5 honor students, so set up one indicator for each of them. Take a single honor student. In a random return, their own quiz is equally likely to land with any of the 20 students, so the chance they get it back is 	frac{1}{20}. Let Xi = 1 if honor student i gets their own quiz. Then E[Xi] = 	frac{1}{20}. Summing over just the 5 honor students, X = X1 + \cdots + X5, and by linearity E[X] = 5 	imes \frac{1}{20} = \frac{5}{20} = \frac14 = 0.25. Because we tracked a subset rather than all 20 students, the answer is below 1. The full class still averages 20 	imes 	frac{1…

Quizzes returned at random: how many honor students?

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