TITLE: Expected time to roll all 1 through 6 on a die
QUESTION [123 upvotes]: What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die?  The order in which the numbers appear does not matter.
I had this questions explained to me by a professor (not math professor), but it was not clear in the explanation.  We were given the answer $(1-(\frac56)^n)^6 = .5$ or $n = 12.152$
Can someone please explain this to me, possibly with a link to a general topic?

REPLY [5 votes]: Associate a success with each number appearing that has not appeared before. Let $X_i$ be the number of trials between the $i^{th}$ success and the $(i + 1)^{st}$ success. 
Let $X$ be the random variable representing the total number of trials required for the required event, and $E[X]$ be the required expected value.
Then by linearity of Expectation, we have $E[X] = 1 + \sum_{i=1}^{5}E[X_i]$.
To calculate $E[X_i]$, consider the following,
after receiving $i−1$ different numbers i.e after $i −1$ successes, each subsequent trial has probability $(6 − i)/6$ of getting a number that has not been appeared before. 
Therefore,the random variable $X_i$ is geometric with parameter $p_i = (6−i)/6$, therefore $E[X_i] = 1/p_i = 6/(6-i)$.
It follows that $E[X] = 1 + 6\sum_{i=1}^{5}1/i$.
Hence $E[X]=14.7$.