Solution: 2017 Fall Final - 18

Author: Michiel Smid

Question

You flip a fair red coin once, and you flip a fair blue coin once, independently of each other. Define the random variables

$X = \bigg\{$	$1\ $	if the red coin flip resulted in heads$,$
$X = \bigg\{$	$0\ $	if the red coin flip resulted in tails$,$
$Y = \bigg\{$	$1\ $	if the blue coin flip resulted in heads$,$
$Y = \bigg\{$	$0\ $	if the blue coin flip resulted in tails$,$

and

What is the expected value $\mathbb{E}(Z)$ of the random variable $Z$?

(a)

(b)

1/2

(c)

1/4

(d)

3/4

Let’s find the individual Expected probabilities for Z

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

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

$ \mathbb{E}(X) = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} $

$ \mathbb{E}(X) = \frac{3}{4} $