Book and misprints - Probability

In some book, $$250$$ misprints are randomly and independently distributed on $$500$$ pages. What is the probability that there are no misprints on the first three pages?

I'm not sure how to solve this. Is using binomial distribution and adding the probabilities $$P_1, P_2, P_3$$ where $$P_1$$ is the probability that there is no misprint on the first page, $$P_2$$ on the second page and so on a good approach? How would you solve this problem? Any help would be much appreciated.

• Number of misprints on a page is often modelled as having a Poisson distribution. Similar question: math.stackexchange.com/q/3679879/321264. Commented May 19, 2020 at 7:31

It's not binomial. That we be correct if it were a case of sampling without replacement, but nothing says we can't have more than one misprint on a page. So a model of the problem is: we draw a number from $$1$$ to $$500$$ at random $$250$$ times. What is the probability that we never draw the number $$1,\ 2\text{ or }3$$?
• Is it $0.14$? I used the Poisson distribution $P(X = 0)=$ $\dfrac{2^0 \cdot e^{-2}}{1} = 0.14$ with $\lambda = 2$ and $x = 0$ (x being the number of errors we're interested in) Commented May 20, 2020 at 7:41
• I meant that the distribution is hypergeometric, so that the probability it $\left(\frac{497}{500}\right)^{250}\approx.22212$. If you want to use the Poisson approximation, then $\lambda=.5$, the average number of misprints per page, and the probability that there are no misprints on the first three pages is $\left(e^{-.5}\right)^3\approx.22313$ Commented May 20, 2020 at 13:19