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Problem Statement: You roll a fair six-sided die until all six numbers have been rolled at least once. What is the expected value of that ?

Can you write different ways to solve this problem ?

Abhisek
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  • https://math.stackexchange.com/questions/28905/expected-time-to-roll-all-1-through-6-on-a-die/28907 – galmeida Jan 05 '19 at 22:18

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Without loss of generality let us focus on one of the numbers, say $1$. The probability that we see $1$ on the $n^{\text{th}}$ attempt given that we did not observe $1$ on the previous $n-1$ tries is given by:

${(\frac{5}{6})}^{n-1} \cdot \frac{1}{6}$

You can use the above to compute the expected value of the number of attempts to make to see $1$. We then multiply the above by $6$ to get the answer we seek.

Srikant
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