Solution: 2022 Winter Final - 18

Author: Michiel Smid

Question

Let $X$ and $Y$ be the results of rolling two $4$-sided dice and let $Z=\max\{X,Y\}$. What is $ E(Z)$?

(a)

$13/4=3.25$

(b)

$25/8=3.125$

(c)

$3$

(d)

$5/2=2.5$

(e)

$7/2=3.5$

Let’s make a table and see what the maximum value is

$ E(Z) = 1 \cdot \Pr(Z=1) + 2 \cdot \Pr(Z=2) + 3 \cdot \Pr(Z=3) + 4 \cdot \Pr(Z=4) $

$ E(Z) = 1 \cdot \frac{1}{16} + 2 \cdot \frac{3}{16} + 3 \cdot \frac{5}{16} + 4 \cdot \frac{7}{16} $

$ E(Z) = \frac{1}{16} + \frac{6}{16} + \frac{15}{16} + \frac{28}{16} $

$ E(Z) = \frac{50}{16} $

$ E(Z) = \frac{25}{8} $