Solution: 2016 Fall Final - 24

Author: Michiel Smid

Question

You repeatedly and independently roll a fair die until the result of the roll is divisible by 3. Define the random variable $X$ to be the number of times you roll the die. For example, if the results of the rolls are 4, 5, 1, 4, 3, then $X=5$.
What is the expected value $\mathbb{E}(X)$ of $X$?

(a)

(b)

(c)

(d)

Solution

Let’s find the probability of rolling a number that is divisible by 3

The numbers that are divisible by 3 are 3 and 6

$ Pr(X=1) = \frac{2}{6} $

We can use the geometric distribution to find the expected value of X

Basically, just do $ \frac{1}{p} $

$ \mathbb{E}(X) = \frac{1}{ \frac{2}{6}} $

$ \mathbb{E}(X) = 3 $