Coupon collector problem
= Coupon collector problem
{wiki}
When independent uniform draws are made from $n$ types, the time $T_n$ until every type has appeared decomposes as a sum of independent geometric waiting times with parameters
$$
1,\frac{n-1}{n},\ldots,\frac1n.
$$
Consequently
$$
\mathbb ET_n=nH_n,
\qquad
\operatorname{var}(T_n)\leq n^2\sum_{j=1}^{\infty}\frac1{j^2},
$$
and Chebyshev's inequality gives $T_n/(n\log n)\to1$ in probability.