Suppose we have an assignment $Y = X$ and that $X \in \{0,1,\dots,255\}$. Using entropy, I want to know how many bits of information passed in the assignment given that $Y > 80$.
If I state that $Y = 80$, I get that all $8$ bits of $X$ flowed into $Y$. On the other hand, if I state that $Y > 80$, I still get $8$ bits of flow, since no matter what the following equation always holds
$$H(X|Y=j) = 0$$
When I calculate entropy, I get a sum of all $H(X|Y=j)$, which is still $0$, as follows
$$H(X|Y=0) p(Y=0) + H(X|Y=1) p(Y=1) + \cdots + H(X|Y=255) p(Y=255) = 0 + 0 + \cdots + 0 = 0$$
How is it possible if we don't know the exact value of $Y$ that the number of bits flowing is the same? I understand that the bits have passed into $Y$ but how do I take into account that fact that $Y$ is not a specific value in the case of $Y > 80$?