Solution: 2014 Winter Final - 23

Author: Michiel Smid

Question

We roll a pair of fair dice repeatedly and independently, and stop as soon as the sum of the numbers for the pair is 7. Define the random variable $X$ to be the number of times we roll the dice. (In one roll, we roll a pair of dice.) What is the expected value of $X$?

(a)

(b)

(c)

(d)

Solution

Let S be the set of all pairs of dice rolls.
The size of S is the number of outcomes when rolling a pair of dice: $ |S| = 6 \times 6 = 36 $
Let A be the set of all pairs of dice rolls that sum to 7.
The set of A is $ { (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) } $
The size of A is the number of outcomes in $A$: $ |A| = 6 $
$ Pr(A) = \frac{|A|}{|S|} = \frac{6}{36} = \frac{1}{6} $

The Geometric Distribution is used to model the number of trials needed to achieve the first success in repeated independent Bernoulli trials.

The expected value of a Geometric Distribution with probability of success $ p $ is given by $ \frac{1}{p} $.

Therefore, the expected value of $ X $ is $ \frac{1}{ \frac{1}{6}} = 6 $.