Calculating Entropy of Dependent Random Variables

Question

So basically I'm trying to answer the following exam problem:

I'm half struggling on H(Z | X) and H(X | Z) and mainly just need confirmation. I know that

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H(Z | X) = -SUM P(Z|X)P(X)logP(Z|X)

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I'm wondering whether the best way of answering is simply calculating P(Z|X) for all possible values and using the above formula. This may seem like an odd question, but it just seems too time consuming (unless I'm doing it totally the wrong way) for such a tiny question.

Just for reference, I make H(Z | X) = 11/12

Thanks in advance!

Answer 1

Like johnniac said, if you know what X is, (e.g. x = 5) there are only two possible values that Z can take(e.g 5 and 6) and they are both equally likely. This means that the uncertainty left about Z when you know X is one bit for any x (e.g. $H(Z|X=5) = H(Y+X|X=5) = H(Y) = 1$).

And since the entropy $H(Z|X)$ is the average uncertainty over all possible x, $H(Z|X) = 1$.