How many people until someone shares YOUR birthday?

Question

Birthdays are independent and uniform over $365$ days. You walk into a room and want someone to share your specific birthday.

How many other people do you need for a better-than-even chance that at least one matches your date? Why is this so different from the famous 23?

Accepted Answer

Use the complement: nobody matches your date. Each person misses it with probability \frac{364}{365}, independently, so with n others P(	ext{no match}) = \left(\frac{364}{365}ight)^n, \qquad P(	ext{at least one match}) = 1 - \left(\frac{364}{365}ight)^n. Set the match probability above 	frac12: \left(\frac{364}{365}ight)^n < 	frac12 \implies n \frac{\ln 0.5}{\ln(364/365)} = \frac{-0.6931}{-0.002743} \approx 252.6, so you need n = 253 people. Why 253, not 23? The classic problem counts all \binom{n}{2} pairs, which grows like n^2, so matches appear fast. Fixing your own date gives on…

How many people until someone shares YOUR birthday?

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