Past exam of the information theory course of the University of Oxford / 2021 / 3 / e

There are two possible messages and three channel uses, givingConditional on either transmitted message, a decoding error occurs exactly when at least two of the three bits are flipped. The discrete memoryless channel makes these flips independent, each with probability $\epsilon =1/10$. Their count therefore has a binomial distribution with parameters $3,\epsilon$. Majority-vote decoding of a repetition code givesInterchanging the two bit values leaves the binary symmetric channel and majority decision unchanged, so $\mathbb{P}(\text{error}\mid W=b)=7/250$ as well. Both the maximum conditional error and the average error, for any message prior, are therefore $0.028$. The probability of correct decoding for each message is $243/250=0.972$. These are finite-block error probabilities; a channel-capacity statement alone would not determine them.