Solution: 2019 Fall Final - 20

Author: Michiel Smid

Question

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

(a)

1/2

(b)

(c)

(d)

1/4

Solution

Let $X_1$ be the amount of dollars you win if the first coin comes up tails: $-1$

There’s a 50/50 chance of it landing tails

$ Pr(X_1 = -1) = \frac{1}{2} $

Let $X_2$ be the amount of dollars you win if the second coin comes up heads: $2$

There’s a 50/50 chance of it landing heads

$ Pr(X_2 = 2) = \frac{1}{2} $

$ E(X) = (-1) \cdot Pr(X_1 = -1) + 2 \cdot Pr(X_2 = 2) $

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

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

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