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The birthday paradox

How many people do you need to have in a room for there to be a 50% chance of two people with a birthday on the same day (but not necessarily the same year)? How many to have a 99% probability of a match?

There only needs to be 23 people in a room for there to be a 50% chance of a shared birthday and only 57 people for there to be a 99% chance! Surprised?

A start to the solution:

What are the chances of having NO matches in the group? If we can find out the probability of having no matches at all (the complementary event), then the probability of finding a match will be 1 minus this probability.

Imagine a room with just one person (A) in it. With A being the only person in the room, it is obvious there are no shared birthdays.

A new person (B) comes to the door. B is only allowed into the room if he or she doesn’t share a birthday with A. What are the chances of that happening? Ignoring February 29, the chances are 364/365 as A has used up one of the birthdays in the year. This means that the chance of having two people in the room who don’t share a birthday is 364-/365. When there are two people, the chance of having a match is the complement, or 1−364/365 = 1/365.

Imagine that C turns up. This person is only allowed into the room if his or her birthday does not match either of the two people already in the room. Since those two people have used up two birthdays from the year, the chance of being able to let C into the room must be 363/365. The chances of having three people in the room without a common birthday must be 364/365 × 363/365. The complement of this, 1– (364/365 × 363/365) then gives the probability of the three people in the room actually having a matching pair of birthdays (or even more than one).

Read a more detailed explanation in Australian Mathematics Teacher 70(1) 2014 journal ‘Desperately seeking birthday mates! Or what maths teachers get up to on Saturday nights!’ 

 Read more in the article ‘The birthday paradox at the World Cup’