Forty guests, any shared birthday

Question

At a dinner with 40 guests, assume birthdays are spread evenly over 365 days and nobody was born on February 29. What is the probability that at least two guests share a birthday?

Accepted Answer

Find the chance that all 40 birthdays are different, then subtract from 1. P(	ext{all 40 different}) = \frac{365}{365}\cdot\frac{364}{365}\cdots\frac{326}{365}. This product works out to about 0.109. Therefore P(	ext{at least one shared birthday}) \approx 1 - 0.109 = 0.891 \approx 89\%. Only 40 people, yet a shared birthday is nearly a sure thing. The reason is the pair count again: \binom{40}{2} = 780 pairs, each a chance to match, and with that many chances a collision is very likely. Takeaway. By 40 people a shared birthday is about 89\% likely, close to certain. Past the 23-person break…

Forty guests, any shared birthday

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