# Mutual information between two mutually exclusive, dependent random variables

I have the following question:

Two horses are in a race under the assumption that there cannot be a draw. The probability of Horse A winning is $p$. What is the mutual information between the two random variables $X$ and $Y$ that model the race outcome (win or lose) for each horse.

And have answered it using the following equations (apologies for the screenshot):

I have calculated that there is no mutual information between the two random variables, but surely there cannot be none if one horse losing depends on the other winning, and vice versa?

Even without the calculation, it should be clear that $H(Y|X)=0$, because given $X$, you know $Y$ completely.
$Pr[Y=y|X=1]$ is zero for $y=1$ and one for $y=0$.
$Pr[Y=y|X=0]$ is zero for $y=0$ and one for $y=1$.