What is X? Is it a constant or random variable? What is its distribution (if its a random variable)?
I think the issue might be this: The problem asks you to find the expected values of Z and their probabilities, not the expected value of Z. This is probably a poor choice of words. It would be better to call them possible values or something like that.
It is clear that the possible values of Z are the same as those of X, since multiplying with Y means multiplying by either 0 or 1. Can you find the new probabilities?