Expected same-month pairs among thirty students

Question

Each of 30 students independently picks a random month (one of 12, equally likely) to present a report. On average, how many pairs of students pick the same month?

Accepted Answer

Linearity of expectation lets us just add up per-pair chances. Number of pairs of students: \binom{30}{2} = \frac{30 \cdot 29}{2} = 435 	ext{ pairs}. Each pair picks the same month with probability 1/12. So E[	ext{same-month pairs}] = 435 	imes \frac{1}{12} = 36.25. With only 12 months and 30 students, same-month pairs are abundant: about 36 on average. (This is far more than the number of students, which is fine, since we are counting pairs, and each student belongs to many pairs.) Takeaway. Coarse buckets produce many collisions. With 12 months and 30 students the expected same-month pai…

Expected same-month pairs among thirty students

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