Shared birthday among thirty party guests

Question

Thirty guests arrive at a party. Assume each birthday is equally likely and independent across 365 days. What is the probability that at least two guests share a birthday?

Accepted Answer

Directly listing all the ways two or more people could match is painful. Instead compute the reverse, all thirty birthdays distinct, and subtract from 1. Seat the guests one at a time, each needing a birthday not yet used: - Guest 1: probability 365/365. - Guest 2: must miss 1 taken day, probability 364/365. - Guest 3: must miss 2 taken days, probability 363/365. - continuing down to guest 30, who must miss 29 taken days, probability 336/365. Multiplying all thirty factors: P(	ext{all distinct}) = \frac{365}{365}\cdot\frac{364}{365}\cdots\frac{336}{365} \approx 0.294. So P(	ext{at least two…

Shared birthday among thirty party guests

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