Using Poisson Distribution Method

Question

Q) The question is as follows: suppose that a book of 200 pages contains 20 printing mistakes.Assume that are errors are randomly distributed throughout the book and x , the number of errors per page has a poisson distribution.Find the probability that 30 pages selected at random will be free of errors.

I could only come with the equation

$$\frac{e^{-20}20^{30}}{30!}.$$

But it is giving wrong answer.What is the solution to this problem?

Answer 1

The rate is $1/10$ per page, so $\lambda=30/10=3$ for $30$ pages. So you want a Poisson with $\lambda=3$. And then $Pr(x=0)=\frac{e^{-3}}{0!}10^{0}=e^{-3}$

Answer 2

The rate $\lambda$ can be seen as equal to 1 for 10 pages. We have 20 such 'stacks' of pages. The probability of success, is $e^{-1}$. Now use Binomial distribution for $k=3, n=k, p = e^{-1}$, so the probability of 3 successes in 3 trials is $\binom{3}{3} e^{-3}(1-e^{-1})^0 = e^{-3}$.