probability - How many throws of a die until every possible result is obtained?

A die is thrown until every possible result (i.e., every integer from 1 to 6) is obtained. Find the expected value of the number of throws.

How do I do that? I understand that probability for the single result is $\{1, 5/6, \ldots , 1/6\}$, but what about the expected value?

Answer

This is a very popular problem. I learned it as the "collector's problem".

Essentially, you want to model rolling a die until a new face is shown
as a geometric distribution with $p_k = \frac{7-k}{6}$ where $k = 1,\dotsc,6$ is the number of faces you have seen. So, if $X_k$ denotes rolling until you see $k$th different face, then $X_k\sim\text{Geom}(p_k)$ on $\{1,2,3,\dots\}$. It follows that $X = X_1+\dotsb+X_6$ is the number of rolls until you have seen all six faces. Then

$$E[X] = E[X_1]+E[X_2]+\dotsb+E[X_6] = \frac{6}{6}+\frac{6}{5}+\dotsb+\frac{6}{1}=14.7.$$