# probability of repeated events

I have a website and I want to calculate the probability of clicks on the ads.

Let the probability that each user clicks on a link be p (something like 1%)

if we have totally N users, What is the formula that computes the probability of exactly n clicks? of course we have

0< = n <= N

Each users can click only once

Consider each user as a trial. For every trial you have two outcomes, they are success (clicks on ad) and failure (does not click on ad). $$P[success] = p$$ and $$P[failure] = 1-p$$.

The total number of ways in which $$n$$ users can be selected from $$N$$ users is $$\binom{N}{n}$$. So, the probability of exactly $$n$$ clicks is $$\binom{N}{n}*p^n*(1-p)^m$$ where $$m = N-n$$, as the clicks are independent and exactly $$n$$ clicks mean exactly $$N-n$$ 'not clicks'.

• And this is called the binomial distribution.
– ken
Oct 11, 2018 at 20:51
• Yes @Ken sure it is!. Oct 12, 2018 at 14:35
• Thanks @naive, is there any function to get (N n) for large Ns. (calculating N! for large Ns is impossible) an approximate function would be sufficient. Oct 27, 2018 at 18:28
• For large $N$ the binomial random variable can be described by a Normal distribution. Oct 28, 2018 at 7:28