Question

A group of thirty people is selected at random. What is the probability that at least two of them will have the same birthday?

Answer 1

Roughly `0.706`

Explanation:

I had to look up the methodology on this - it's here.

I'm going to approach this problem by asking that the probability is that no two people have the same birthday (the probability that at least 2 people share the same birthday is the same as 1 - the probability that no two people share the same birthday) - it makes the math easier.

To do this, we start off by saying that the first person we pick has a birthday and it can be on any day of the year (I'm going to use 365 day years for this) so we can say:

`365/365`

The next person, person 2, can have a birthday on any day of the year except for the one that person 1 has, so s/he has `364/365` to have one in, so that means that we say the odds of the two of them having different birthdays is:

`365/365xx364/365`

And we can continue in like vein. We end up with a formula for n people having different birthdays as:

`(365/365)((365-1)/365)((365-2)/365)...((365-n+1)/365)` which is the same as:

`P_(365,30)/(365^30)=(365!)/((365-30)!(365^30)`

Normally I'd work this out manually, but I'll use a couple of online tools: a permutation calculator and google calculator.

All of this comes to roughly `0.294`

So the odds of at least two people having the same birthday is `1-0.294=0.706`