Question: 2018 Fall Final - 18

A red box contains the numbers 0, 1, and 2, and a blue box also contains the numbers 0, 1, and 2. You choose a uniformly random element from the red box and a uniformly random element from the blue box; these two choices are independent of each other. Define the random variables

• X = the number you choose from the red box,
• Y = the number you choose from the blue box,
• Z = $\max(X, Y)$.

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