There are 50 misprints in a book which has 250 pages, find the probability that page 100 has no misprints? (Use theoretically correct distribution)

Question

My question is where this should be modelled as a binomial distribution problem or a Poisson distribution problem.

Any hint/advice helps, thanks in advance!

Answer 1

The answer depends on whether what perspective we take on the problem:

Case 1: Binomial - We know upfront that there are 50 misprints.

Since multiple errors can appear on the same page, we can first count the number of ways the misprints can be placed using stars and bars.

This gives us $299\choose50$ for the total number of ways for the misprints to be distributed. To count the number of ways where page 100 has no misprints, we assume the book has 249 pages, giving us $298\choose50$. So, our answer is $$\frac{298\choose50}{299\choose50}=\frac{249}{299}\sim\color{red}{0.8238}$$

Case 2: Poisson - The misprints happened randomly, but it just so happened that there were 50 of them

In this case, we consider each misprint as an independent event, and consider the probability of it not occurring on a given page.

This gives us $$\bigg(\frac{249}{250}\bigg)^{50}\sim\color{red}{0.8184}$$

Answer 2

If we assign each misprint a value then P(100 does not have the $n^{th}$ misprint) $= \frac {249}{250} $

Since each event is independent,

P(100 does not have any misprint) $=(\frac {249}{250})^{50} = 0.81840245067$