Solution: 2019 Winter Final - 23

Author: Michiel Smid

Question

You are given a fair red die and a fair blue die. Consider the following experiment:

Experiment: Roll each die once and take the sum of the two rolls.

You repeat this experiment until the sum of the two rolls is equal to 7.
Consider the random variable

X = the number of times you do the experiment.

(This value $X$ includes the experiment in which the sum is 7 for the first time.)
What is the expected value $\mathbb{E}(X)$ of the random variable $X$?

(a)

(b)

(c)

(d)

Solution

Let’s draw out a sum table

We can see that sum of 7 occurs 6 times out of the 36 possible outcomes.

Therefore, the probability of the sum being 7 is $\frac{6}{36} = \frac{1}{6}$.

Now, we use geometric distribution to find the expected value of the number of rolls until the sum is 7.

The expected value of a geometric distribution is $\frac{1}{p}$, where $p$ is the probability of success.

Therefore, the expected value of the number of rolls until the sum is 7 is $\frac{1}{\frac{1}{6}} = 6$.