Write the expression for $\mathbb{E}(Y|X)$

Question

I have a confusion regarding how to go about solving the following question:

Suppose you are invited to play a game where your earnings are given by multiplying the outcome of rolling a fair die ($Z$) with tossing a fair coin ($X$). You have to pay $\$5$ each time you want to play this game.

Let $Y$ = net earnings, $Z$ = outcome of dice ($1,2,\ldots,6$), $X$ = outcome of coin ($X$ equals $1$ if heads and $2$ if tails)

• Write $Y$ in terms of $X$ and $Z$

This comes out as $Y = XZ-5$ (I think)

• Write the expression for $\mathbb{E}(Y|X)$. What is the expected earnings if
  we know the coin landed tail

• What is your expected earning from playing this game? Answer
  without making a probability distribution for $Y$.

Answer 1

You just have to use the fact that dice roll and coin toss are independent, after that (and the fact that for independent variables $\mathbb{E}(XY) = \mathbb{E}(X) \cdot \mathbb{E}(Y)$) it's rather trivial.