What is the probability distribution for this question?

Question

I tried to find the probability distribution for this problem and then calculate the expectation of $n$. I suppose the total states of our problem would be $52^n$ and our desired states would be $52$ combination of n with permutation $52!$ for distinct cards and our probability function will be $p(n)=\frac{n!}{(n-52)!(52^n)}$ with $n = 52,53, ....$ But the summation over $p(n)$ is not $1$. I don't know what is wrong with this solution. Can anybody help? Thank you so much

Answer 1

This is the Coupon Collector's Problem

Answer 2

Have a look at $p(53)$: $$p\left(53\right)=\frac{52}{52}\frac{\color{red}1}{\color{red}{52}}\frac{51}{52}\frac{50}{52}\cdots\frac{1}{52}+\frac{52}{52}\frac{51}{52}\frac{\color{red}2}{\color{red}{52}}\frac{50}{52}\cdots\frac{1}{52}+\cdots+\frac{52}{52}\frac{51}{52}\frac{50}{52}\cdots\frac{2}{52}\frac{\color{red}{51}}{\color{red}{52}}\frac{1}{52}$$

The colored factors correspond with the unique non-hit.

This is not the expression for $p(53)$ that you suggest.