How few people for two birthdays in the same week?

Question

Call two people a near-match if their birthdays are within $7$ days of each other. 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

First find the chance a single pair near-matches, then run the birthday-problem logic with that larger probability. Fix one person's birthday. The other near-matches if their day lands within 7 on either side, that is the day itself plus 7 before and 7 after, giving 15 of the 365 days: p = \frac{15}{365}. With n people, 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{15}{365} \approx \ln 2 = 0.693 \implies \binom{n}{2} \approx 16.9 \implies n(n-1) \approx 34 \implies n \approx 6.3. Roundin…

How few people for two birthdays in the same week?

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