Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 3I / b / i / Solution

With rows indexed by outputs $A,B,\star$ and columns by inputs $A,B$, the channel matrix isThis is a binary erasure channel with erasure probability $1/2$. Conditional on no erasure, the input is recovered exactly, whereas an erasure supplies no information. Thuswhich is maximized by the uniform input distribution. Therefore