This week's CR4 Challenge Question:
How many people must be in a room in order for the probability to be greater than 1/2 that at least two of them have the same birthday? (By "same birthday", we mean the same day of the year; the year may differ.) Ignore leap years.
Thanks to Maths_Physics_Maniac for this question!
Answer:
Given n people, the probability, Pn, that there is not a common birthday among them is
The first factor is the probability that two given people do not have the same birthday. The second factor is the probability that a third person does not have a birthday in common with either of the first two. This continues until the last factor is the probability that the nth person does not have a birthday in common with any of the other n  1 people.
We want Pn < 1/2. If we simply multiply out the above product with successive values of n, we find that P22 = 0.524, and P23 = 0.493
Therefore, there must be at least 23 people in a room in order for the odds to favor at least two of them having the same birthday.

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