Chapter 15.4, Problem 10P

What is the probability that you and a friend have different birthdays? (For simplicity, let a year have 365 days.) What is the probability that three people have three different birthdays? Show that the probability that n people have n different birthdays is $p=\left(1-\frac{1}{365}\right)\left(1-\frac{2}{365}\right)\left(1-\frac{3}{365}\right)\cdots \left(1-\frac{n-1}{365}\right)$

Estimate this for $n\ll 365$ by calculating ln $p$ [recall that $\ln (1+x)$ is approximately $x$ for $x\ll 1].$ Find

he smallest (integral) $n$ for which $p<\frac{1}{2}.$ Hence, show that for a group of 23 people or more, the probability is greater than $\frac{1}{2}$ that two of them have the same birthday. (Try it with a group of friends or a list of people such as the presidents of the United States.)