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Expected birthday pairs

ProbabilityMedium

In a room there are $20$ people with independent birthdays, each uniform over $365$ days (ignore leap days). What is the expected number of pairs of people who share a birthday?

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Hint

Count the pairs and use linearity of expectation with one indicator per pair.

Solution

The number of pairs is $20 \times \frac{19}{2} = 190$ .
Each pair shares a birthday with probability $\frac{1}{365}$ .
By linearity the expected number of matching pairs is $\frac{190}{365}$ .
Dividing numerator and denominator by $5$ gives $\frac{38}{73}$ .

Answer: $\frac{38}{73}$

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