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The simplest way to prove the above is to consider the probability that no two people in a room share the same birthday, i.e. that everybody has a different birthday:

If there are 2 people in a room, the probability that they will have different birthdays is 364/365

If there are 3 people in a room, the probability that they will all have different birthdays is (364/365) x (363/365)

If there are 4 people in a room, the probability that they will all have different birthdays is (364/365) x (363/365) x (362/365)

It follows that if there are 23 people in a room the probability that they will all have different birthdays is (364/365) x (363/365) x (362/365) x (361/365) … (343/365)

This can be expressed as 364!/(342! x 365^{22}) = 0.4927

0.4927 is the probability that no two people in a room of 23 people share the same birthday. Therefore the probability that at least two people in a room of 23 people share the same birthday is (1 – 0.4927 = 0.5073)

0.5073 is greater than 1/2, which means that it is more likely than not.

Q.E.D.

NB For simplicity we have assumed that there are 365 days in every year. It has also been assumed that it is equally likely for a random person to have a birthday on any of the 365 days in a year. In reality some dates are more common for people's birthdays than other, which makes it more that two people share the same birthday.

For those unfamiliar with mathematical notation, 5! = 5 x 4 x 3 x 2 x 1.

In the above, "x" is used to mean "multiplied by".

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