Birthday Paradox

Birthday paradox is an ancient mathematical problem. There are a great deal of different variations from it but all in all the question is always the same: How many people you need to have so it's more likely that two of them have the same birthday.

Many guys are a little surprised when they notice someones have the same birthday, after all there are 365 days in a year. But actually the amount of people you need is quite small.

It's easy to show that already 23-member group is more likely that at least two people have the same birthday, as all different. This can be calculated by how likely it is that everyone is born on a different days as following:

Quote by wiki

It is easier to first calculate the probability p(n) that all n birthdays are different. If n > 365, by the pigeonhole principle this probability is 0. On the other hand, if n is smaller of equal to 365, it is:



because the second person cannot have the same birthday as the first (364/365), the third cannot have the same birthday as the first two (363/365), etc.

The event of at least two of the n persons having the same birthday is complementary to all n birthdays being different. Therefore, its probability p(n) is:



This probability surpasses 1/2 for n = 23 (with value about 50.7%).

So maybe that's the reason why I happily wish sincere happy birthday to Flu aka. Firpo, who is somewhere in Amsterdam at the moment, and to Meez, who is somewhere in Leedzzzfest celebrating I hope.

Happy Birthday both!