Consider the following problem:
What is $I(X;Y)$ where $X$ is the outcome of a roll of a fair 6-sided die and $Y$ is whether the outcome of THAT SAME ROLL was even or odd?
Intuitively, I thought that $I(X;Y) = H(X) + H(Y) - H(X, Y)$ - that is, when given even or odd, we cut $X$ in half (so we know the number was either 1, 3, 5 or 2, 4, 6). However, I'm slightly confused (as my professor argued that the correct answer was 1).
His idea was that if I'm given the number rolled, then all of the even/odd information is contained within, so $I(X;Y) = 1$.
Maybe I don't understand mutual information correctly, an explanation would be appreciated.