Solution: 2015 Winter Final - 19

Author: Michiel Smid

Question

I flip a fair coin, independently, $n$ times. For each heads, you win \$3, whereas for each tails, you lose \$1. Define the random variable $X$ to be the amount of dollars that you win. What is the expected value of $X$?

(a)

$n$

(b)

$n/2$

(c)

$3n/2$

(d)

$2n$

Solution

Let A be the event that you win \$3 by flipping a tails
$ Pr(A) = \frac{1}{2} $
Let B be the event that you lose \$1 by flipping a heads
$ Pr(B) = \frac{1}{2} $

$ \mathbb{E}(X) = 3 \cdot Pr(A) - 1 \cdot Pr(B) $

$ \mathbb{E}(X) = 3 \cdot \frac{1}{2} + (-1) \cdot \frac{1}{2} $

$ \mathbb{E}(X) = ( \frac{3}{2} - \frac{1}{2} )n $

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

$ \mathbb{E}(X) = n $