Madness In The Middle Kingdom

If you have 23 people in your class there is a slightly greater than 50% chance that 2 will share the same birthday. If there are 30 people then there is a 71% chance 2 will share the same birthday and if you have 60 people there is a 99.4% chance - an almost certainty.

Below is a graph that shows the probability plotted against the number of people in the group

I have 3 math classes of 22, 20 and 23 students (started with 27,27,27 so I must not be that popular). I tried this with them and in 2 of 3 classes two people matched birthdays.

Well .. this one was a bit more challenging to solve than the previous problems so if you are interested then take a look at the solution. It is a bit difficult but you might learn something if you are interested.

In probability when we cannot solve something directly we can look at the complementary event which is essentially the opposite event. The probability of an event and the probablity of its complement add up to be 1.
Example - Event: It is raining Complement: It is not raining
The probability that it is raining or not raining is 1 since it must be doing one or the other.
P(raining) + P(not raining) = 1 OR P(raining) = 1 - P(not raining)

This is how we solve the birthday problem.
P(a match among N people) = 1 - P(no match among N people)

To calculate the complement probability imagine there are 3 people in the class. The first person can have any birthday. The second person’s birthday has to be different. There are 364 different days to choose from, so the chance that two people have different birthdays is 364/365. That leaves 363 birthdays out of 365 open for the third person.

To find the probability that both the second person and the third person will have different birthdays, we have to multiply:

(365/365) * (364/365) * (363/365)

A formula for the probability that n people have different birthdays (no matches) is

((365-1)/365) * ((365-2)/365) * ((365-3)/365) * . . . * ((365-n+1)/365).

Or the probability that out of N people 2 will have the same birthday is

1 - ((365-1)/365) * ((365-2)/365) * ((365-3)/365) * . . . * ((365-n+1)/365)

Throw in n = 23 and it is just bigger than 50%