How few people for two near-birthdays?

Question

Call two people a "near-match" if their birthdays are within $1$ day of each other (the same day, or one day apart). Birthdays are independent and uniform over $365$ days.

What is the smallest number of people for which a near-match is more likely than not?

Accepted Answer

The trick is to figure out the chance a single pair is a near-match, then treat the problem like an ordinary birthday problem with that larger per-pair probability. Fix one person's birthday. The other person near-matches if their birthday falls on that day, the day before, or the day after, that is 3 of the 365 days. So p = \frac{3}{365}. Now with n people there are \binom{n}{2} pairs, and the chance of no near-match anywhere is approximately P(	ext{no near-match}) \approx e^{-\binom{n}{2}\,p}. Set this to 	frac12: \binom{n}{2}\cdot\frac{3}{365} \approx \ln 2 = 0.693 \implies \binom{n}{2} …

How few people for two near-birthdays?

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