Mutual Information in a Binary Erasure Channel Imagine a Binary Erasure Channel as depicted on Wikipedia.

One equation describing the mutual information is:

$$
\begin{align*}I(x;y) 
 &amp;= H(x) - H(x|y) \\
 &amp;= H(x) - p(y=0) \cdot 0 - p(y=?) \cdot H(x) -p(y=1)\cdot 0\end{align*}$$

Why is it "$p(y=?) \cdot H(x)$" and not "$p(y=?) \cdot H(x|y=?)$"?
If $y = ?$ then $H(x) = H(x|y=?)$ since $y = ?$ gives you no information about $x$.