Question

How many times would you expect to roll a fair die before all $6$sides appeared at least once?

Short answer

The expected number is$14.7$

Step-by-step solution

Given Information

we have that random variable $I_j\in {\mathrm{\mathbb{N}}}_0$has Geometric distribution with the parameter of success $\frac{6-j+1}{6}$. The total number of needed throws is $N=\sum _j{I}_j$.

Explanation

Therefore, the expected number is,

$E\left(N\right)=\sum _{j}E\left(I_j\right)=\sum _j\frac{1}{\frac{6-j+1}{6}}=\sum _j\frac{6}{6-j+1}=14.7$

Final Answer

The expected number is$14.7$