Shared birthdays among party guests
Programming Snapshot – Probability
At a party with 23 guests, having two guests with the same birthday in more than 50 percent of cases may sound fairly unlikely to amateur mathematicians. Armed with statistical methods, party animal Mike Schilli sets out to prove this claim.
The problem depends on the exact wording. Nobody can expect to go to a party with 23 people and meet someone with the same date of birth with 50 percent probability. The unexpected result comes about by the fact that n guests are compared with each other (i.e., each with (n – 1) other guests). It is much more likely that two random guests will be born in the same month and on the same day (the year is not considered) than if you only compare your own birthday with that of (n – 1) guests [1].
Bottom Up
At a party with only two guests, what is the probability of both celebrating their birthday on the same day? Assuming a year to be 365 days for the sake of ease, without taking into account seasonal birth fluctuations or special cases such as twin parties, this occurs in one in 365 cases. Conversely, the probability that both guests have birthdays on different days is 364 in 365.
If another guest joins the pair, the probability that no one in the room is celebrating their birthday on the same day is the coincidence of two independent events: The first event, which we just calculated to occur with the probability 364/365, and a second event, where the added person does not share a birthday with the first or the second person and can thus celebrate a birthday on only 363 of 365 days.
[...]
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