Solution: 2016 Fall Final - 21

Author: Michiel Smid

Question

Consider a coin that comes up heads with probability 1/7 and comes up tails with probability 6/7. You flip this coin once. If it comes up heads, you win \$5. If it comes up tails, you win \$2.
What is the expected value $\mathbb{E}(X)$ of $X$?

(a)

17/7

(b)

32/7

(c)

7/2

(d)

7/17

Solution

Let's determine X=5
The probability of getting heads is $ \frac{1}{7} $
$ Pr(X=5) = \frac{1}{7} $
Let's determine X=2
The probability of getting tails is $ \frac{6}{7} $
$ Pr(X=2) = \frac{6}{7} $

$ \mathbb{E}(X) = 5 \cdot Pr(X=5) + 2 \cdot Pr(X=2) $

$ \mathbb{E}(X) = 5 \cdot \frac{1}{7} + 2 \cdot \frac{6}{7} $

$ \mathbb{E}(X) = \frac{5}{7} + \frac{12}{7} $

$ \mathbb{E}(X) = \frac{17}{7} $