Birthday Problem inverted - probability of only one birthday on a given date.

Question

Given a group of 250 people, what is the probability that exactly one person will have a birthday on a particular chosen date? There's lots of birthday problem questions on this site but I couldn't find any with this particular twist. Am at a loss and would appreciate any help, hints, or guidance. Thanks!

Answer 1

The probability that person $a$ has a birthday on the date and all others don't is $\frac 1{365}*(\frac{364}{365})^{249}$. Those events are independent so the odds that exactly one person has that birthday are $250*\frac 1{365}*(\frac{364}{365})^{249}\approx .34$.

Answer 2

The number of people with birthday on (say April 1) is binomial distributed, with parameters $n=250$ and $p=1/365$, so your probability is $\binom{250}{1} (1/365)^1 (364/365)^{249}$.