Solution: 2015 Winter Final - 24

Author: Michiel Smid

Question

You repeatedly, and independently, roll two fair dice, until the sum of the values of the two dice is equal to 12. Define the random variable $X$ to be the number of times you roll the pair of dice. What is the expected value of $X$?

(a)

(b)

(c)

(d)

Solution

There are 36 possible ordered pairs of dice rolls.

Let $X_i$ be 1 if the sum of the values of the two dice is 12 and 0 otherwise.

The probability of getting a sum of 12 is $ \frac{1}{36} $ because the only way to get a sum of 12 is to roll a 6 and a 6.

$ Pr(X=1) = \frac{1}{36} $

To find the expected value, we divide 1 by the probability of getting a sum of 12.

$ E(X) = \frac{1}{ \frac{1}{36}} $

$ E(X) = 36 $