Incremental information in a twofold binary repetition code
= Incremental information in a twofold binary repetition code
Send a uniform bit twice through independent binary symmetric channels of crossover probability $p$. The first received bit supplies $1-h_2(p)$ bits. Since the received bits disagree with probability $2p(1-p)$, the additional information in the second bit is
$$
h_2(2p(1-p))-h_2(p).
$$