TITLE: Probability - Book with typos problem
QUESTION [5 upvotes]: In a book, $250$ printing errors are randomly and independently distributed on $500$ pages. What is the probability that there will be at least three printing errors on page $317$?

Is binomial distribution a good way to tackle this problem? My second idea was to use Poisson distribution to approximate the probability. What is the better approach? Any help would be much appreciated.

REPLY [2 votes]: From my perspective, the idea of using binomial distribution can be possible.
For example, 
$$P(\text{only one error on page 317})={250 \choose 1}\times(\frac {1}{500})\times(\frac {499}{500})^{249}$$
Here's my calculation: 
Let X be the number of printing errors on page 317. $X\sim Bin(250,\frac {1}{500})$
$$P(X\geq 3)=1-P(X\lt3)=1-(P(X=0)+P(X=1)+P(X=2))$$
$$P(X\geq 3)=1-((\frac {499}{500})^{250}+{250 \choose 1}\times(\frac {1}{500})\times(\frac {499}{500})^{249}+{250 \choose 2}\times(\frac {1}{500})^{2}\times(\frac {499}{500})^{248})\approx 0.0143\text{ (corr. to 4 d.p.)}$$