 Solution

Let $X\in \{0,1\}$ denote whether $00$ or $11$ was sent, and writewhere $E_1,E_2$ are independent Bernoulli variables of parameter $p$. Because $X$ is uniform, $Y_1$ is uniform, while $H(Y_1\mid X)=h_2(p)$. Thus the mutual information in the first received digit is

By the chain rule for mutual information, the extra information in the second digit is $I(X;Y_2\mid Y_1)$. Conditional on $X$, the second channel error is independent of the first output, soMoreover,Given either value of $Y_1$, the second output therefore differs from it with probability $2p(1-p)$, andConsequently the incremental information in a twofold binary repetition code isbits.