Question: 2019 Winter Final - 19

A red box contains the numbers 0, 1, and 2; a blue box contains the numbers 0 and 1; and a green box contains the numbers 1 and 2. Consider the following two steps: Step 1: Choose a uniformly random number from the red box, and denote it by $x$.
Step 2:

• If $x = 0$ or $x = 2$, then choose a uniformly random number from the green box, and denote it by $y$.
• Otherwise (i.e., if $x = 1$), choose a uniformly random number from the blue box, and denote it by $y$.

Consider the random variables

• X = the number $x$ you choose in Step 1,
• Y = the number $y$ you choose in Step 2,
• Z = $\max(X,Y).$

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