= Solution
With equally likely codewords, the prior factor in the ideal-observer criterion is constant, so it agrees with maximum <likelihood>. On the <binary symmetric channel>,
$$
p(y\mid x)=p^{d_H(x,y)}(1-p)^{n-d_H(x,y)}.
$$
For $0<p<1/2$ this is strictly decreasing in <Hamming distance>, so the <likelihood> maximizers are precisely the nearest codewords. With the same tie convention all three decoders therefore agree. At $p=0$ every possible received word equals its transmitted codeword, and the rules agree on all positive-probability outputs; outputs having zero probability need no posterior assignment.
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