I'm trying to improve my probability theory and came across the following question which I am struggling with. Suppose there is a 500 page book with 500 misprints in it. What is the probability that a given page will have two or more misprints?
I want to give an approximation of this using that the binomial distribution is approximated by the Poisson distribution. In this case $n = 500$ and the probability of a misprint on any page is $p = 1/500$. So the probability of there being exactly $k$ misprints on a given page is approximately
$$ \frac{1}{k!}e^{-1} $$
thus the probability of two or more misprints on the same page is approximately given by
$$ 1 - e^{-1} - e^{-1} = 1-2e^{-1}. $$
However, the book I am using claims the answer is $1- \frac{5}{2}e^{-1}$. I am not sure where this fraction is coming from. Is this claimed answer itself a misprint?
Any help is appreciated. Thanks!