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Solution: Birthday Problem – in a nutshell.

April 3, 2019

Solution to Exercise

In a group of 24 people, there are in fact 276 pairs of people to choose from.  Therefore, a group of 24 people generates 276 chances, each of size 1/365, of having at least two people in the group sharing the same birthday.

In a group of 24 people, there are, according to the standard formula, 24C2 pairs of people (called 24 Choose 2) pairs of people.

Generally, the number of ways k things can be chosen from n is:

n C k = n! / (n-k)! k!

Here n! (n factorial) is n x n-1 x n-2 … down to 1. Similarly for k!

Thus, 24C2 = 24! / 22! 2! = 24 x 23 / 2 = 276

These chances have some overlap: if A and B have a common birthday, and A and C have a common birthday, then inevitably so do B and C. So the probability that at least two people in the group of 24 do not share a birthday is:

(364/365)276 = 0.469

The odds that at least two of the 24 people share the same birthday = 1 – 0.469 = 0.531 = 53.1%

From → Solutions

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