Why twenty-three people give even odds

Question

There are 23 people in a room, birthdays independent and uniform over 365 days. On average, how many pairs of people share a birthday? (The number you get explains the famous even-odds result.)

Accepted Answer

By linearity of expectation, add the per-pair match chances. Number of pairs among 23 people: \binom{23}{2} = \frac{23 \cdot 22}{2} = 253 	ext{ pairs}. Each pair shares a birthday with probability 1/365, so E[	ext{matching pairs}] = 253 	imes \frac{1}{365} \approx 0.693. That value is no coincidence: 0.693 \approx \ln 2. When the expected number of matches is about \ln 2, the probability of at least one match is close to 1 - e^{-\ln 2} = 1 - 	frac12 = 	frac12. That is exactly why 23 people give roughly even odds of a shared birthday. Takeaway. The expected number of matching pairs at n =…

Why twenty-three people give even odds

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