What is the formula for the birthday problem?
I can't seem to find JUST the formula. I don't care how it works at this point, I just want to know what to do. Can someone please just tell me? Most websites seem to be a convoluted mess (at least to me).
The usual form of the Birthday Problem is: How many do you need in a room to have an evens or higher chance that 2 or more share a birthday.
The solution is $1-P(\text{everybody has a different birthday})$. Calculating that is straight forward conditional probability but it is a mess.
We have our first person. The second person has a $\frac{364}{365}$ chance of having a different birthday. The third person has a $\frac{363}{365}$ chance of having a unique birthday etc.
$$P(\text{everybody different})=\frac{364}{365}*\frac{363}{365}*...*\frac{366-n}{365}=\frac{364!}{(365-n)!365^{n-1}}=\frac{365!}{(365-n)!365^{n}}$$
So your problem is the smallest integer n such that $1-\frac{365!}{(365-n)!365^{n}}>0.5$ which is 23.