# Different ways to calculate the expected value of getting all faces of a 6 sided die atleast once

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.