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Partial spoiler

Posted By: Timothy Chow
Date: Tuesday, 4 May 2010, at 9:05 p.m.

In Response To: Non BG Related Probability Problem - for fun (John McDonald)

The way I like to explain the birthday paradox is as follows. Given any pair of people, the probability that they have the same birthday is (about) 1/365. If there are n people, then there are about n2/2 pairs of people to choose from (the exact number of pairs is n(n – 1)/2 but n2/2 is close). So on average I should expect (n2/2)*(1/365) = n2/730 birthday coincidences among n people. If I'm looking for the 50/50 threshold then this should happen approximately when n2/730 = 1, i.e., when n is around the square root of 730.

This argument isn't quite right because, as illustrated by the roulette problem I posed recently, the 50% probability threshold isn't the same as the average value (median is not the same as mean) but it gets you in the right ballpark and makes the answer less baffling.

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