Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 60 / 2 / ii / a / Solution

The error indicator is a Bernoulli random variable with probabilities $P(Z=1)=p_e$ and $P(Z=0)=1-p_e$. Its Shannon entropy is therefore the binary entropyUse $0{\log }_{2}0=0$, so this expression also covers error probabilities zero and one.