EX 3.39 error page

If a publisher of nontechnical books takes great pains to ensure that its books are free of typographical errors, so that

• the probability of any given page containing at least one such error is .005
• errors are independent from page to page,

Question

What is the probability that one of its 600-page novels will contain exactly one page with errors? At most three pages with errors?

With $S$ denoting a page containing at least one error and $F$ an error-free page, the number $X$ of pages containing at least one error is a binomial rv with $n=600$ and $p=.005$, so $np=3$. We wish

$$\begin{array}{rl}P(X=1)& =b(1;600,0.005)\\ & \approx p(1;3)=\frac{e^{-3}\cdot 3^1}{1!}\\ & =0.14936\end{array}$$

The binomial value is $b\left(1;600,.005\right)=.14899$ , so the approximation is very good.

Similarly, $P\left(X\le 3\right)\approx \sum _{x=0}^{3}p\left(x;3\right)=F\left(3;3\right)=.647$ which to three-decimal-place accuracy is identical to $B\left(3;600,.005\right)$ .