## Birthday Problem (Solution)

More on this <a href=https://en.wikipedia.org/wiki/Birthday_problem>here</a>.

Let’s build up incrementally. The first person has a birthday on some random day. What’s the chance that the second person has the same birthday as the first?

\[1 / 365\]Let’s say he doesn’t share the same birthday as the first (\(P = 1-1/365\)), what’s chance that the third person shares a birthday with one of the first two people?

\[2/365\]To generalize, the chance that the (n+1)th person shares a birthday with first n people, assuming the first n people all have different birthdays, is

\[n/365\]Now to flip it around, the probability that no one shares a birthday out of n people:

\[(1-1/365) (1-2/365) (1-3/365)...(1-(n-1)/365) \\ \prod_{i=1}^{n-1} 1-i/365\]So the probability that at least one two people share a birthday out of n people:

\[P = 1 - \prod_{i=1}^{n-1} 1-i/365\]Now, just find the smallest \(n\) s.t. \(P > 0.5\). For this, I just use a computer and find that when \(n = 23\), \(P \approx 0.507\).