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 ?


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.

  • $\begingroup$ Can you please solve it and find the solution for completeness ? $\endgroup$ – Abhisek Oct 8 '17 at 17:05

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