Poisson Limit of Lottery Winners
A total of
N$ people play a lottery each week. Each person is independently assigned a uniformly random integer from $\{1, 2, \dots, N\}$. One of the $N$ integers is then selected uniformly at random as the winning value. Let $W_N$ be the number of people who win in a given week.
Compute $\displaystyle \lim_{N \to \infty} P(W_N \leq 3)$ to the nearest thousandth.