Solution: 2019 Fall Final - 22

Author: Michiel Smid

Question

You are given a fair 6-sided red die and a fair 6-sided 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 a summation table, dear

As can be seen, there are 6 ways to get a sum of 7: $ { (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) } $

The probability of getting a sum of 7 is $ \frac{6}{36} = \frac{1}{6} $

By the geometric distribution, the expected value is $ \frac{1}{ \frac{1}{6}} = 6 $