Solution: 2015 Fall Final - 23

Author: Michiel Smid

Question

Let $S$ be a uniformly random 2-element subset of $\{1,2,3,4\}$, and let $X$ be the number of elements of $S$ that are even. What is the expected value $\mathbb{E}(X)$ of $X$?

(a)

3/2

(b)

(c)

5/2

(d)

Solution

$ S = {1, 2, 3, 4} $
$ S = {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} $
$ X = 0 $ if $ S = {1, 3} $
$ Pr(X=0) = \frac{1}{6} $
$ X = 1 $ if $ S = {1, 2}, {2, 3}, {1, 4}, {3, 4} $
$ Pr(X=1) = \frac{4}{6} $
$ X = 2 $ if $ S = {2, 4} $
$ Pr(X=2) = \frac{1}{6} $

$ \mathbb{E}(X) = 0 \cdot Pr(X=0) + 1 \cdot Pr(X=1) + 2 \cdot Pr(X=2) $

$ \mathbb{E}(X) = 0 \cdot \frac{1}{6} + 1 \cdot \frac{4}{6} + 2 \cdot \frac{1}{6} $

$ \mathbb{E}(X) = 0 + \frac{4}{6} + \frac{2}{6} $

$ \mathbb{E}(X) = \frac{6}{6} $

$ \mathbb{E}(X) = 1 $