Let $C$ be a binary code of length $n$. Define the following decoding rules: (i) ideal observer, (ii) maximum likelihood, (iii) minimum distance.

Let $p$ denote the probability that a digit is mistransmitted and suppose $p<1 / 2$. Prove that maximum likelihood and minimum distance decoding agree.

Suppose codewords 000 and 111 are sent with probabilities $4 / 5$ and $1 / 5$ respectively with error probability $p=1 / 4$. If we receive 110 , how should it be decoded according to the three decoding rules above?