Best place in a queue for a birthday-match prize

A shop announces a prize for the first customer in the queue whose birthday matches that of somebody ahead of them in the queue. Birthdays are independent and uniform over $365$ days, and you know nobody's birthday. Which position in the queue maximises your chance of winning?

$20$th. At position $n$ you win if the $n-1$ people ahead all have distinct birthdays and yours matches one of them: $P(n)=\tfrac{n-1}{365}\prod_{k=1}^{n-2}\left(1-\tfrac k{365}\right)$. Comparing $P(n+1)/P(n)=\tfrac{n(366-n)}{365(n-1)}$ with 1 shows the maximum is at $n=20$ (about 3.23%).