Solution: 2014 Winter Final - 20

Author: Michiel Smid

Question

I flip two fair and independent coins. If the first coin comes up heads, you win \$1. If the second coin comes up heads, you win \$2. (Thus, if both coins come up heads, you win \$3.) Define the random variable $X$ to be the amount of dollars that you win. What is the expected value of $X$?

(a)

5/2

(b)

3/2

(c)

(d)

Solution

Let A be the event that the first coin comes up heads and you win \$1.
$ Pr(A) = \frac{1}{2} $
Let B be the event that the second coin comes up heads and you win \$2.
$ Pr(B) = \frac{1}{2} $

$ \mathbb{E}(X) = 1 \cdot Pr(A) + 2 \cdot Pr(B) $

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

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

$ \mathbb{E}(X) = \frac{3}{2} $