Q.7.22 - A First Course in Probability

Question

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

Step-by-Step Solution

Answer

The expected number is $14.7$

Step 1: 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 $N=\sum _j{I}_j$.  

Step 2: Explanation

Therefore, the expected number is,

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

Step 3: Final Answer

The expected number is $14.7$