= Solution
Assume the two response labels are distinct and the error probability is the same for each stimulus. Then the stimulus-response map is a <binary symmetric channel>:
$$
\mathbb P(R=r_+\mid S=+)=\mathbb P(R=r_-\mid S=-)=1-P_x,
$$
with the two incorrect conditional probabilities equal to $P_x$. Equal stimulus probabilities give $\mathbb P(R=r_+)=\mathbb P(R=r_-)=1/2$. In bits, the <Shannon entropy> of the response is therefore $H(R)=1$. For either stimulus the response entropy is the <binary entropy>
$$
h_2(P_x)=-P_x\log_2P_x-(1-P_x)\log_2(1-P_x),
$$
with $0\log0=0$. Thus $H_{\mathrm{noise}}=H(R\mid S)=h_2(P_x)$, and the <mutual information> is
$$
\boxed{I(S;R)=1-h_2(P_x)quad\text{bits per presentation}.}
$$
It is maximal, equal to one bit, at $P_x=0$ and also at $P_x=1$. The latter response is always opposite to the ideal label but still specifies the stimulus perfectly after relabelling. It is minimal, equal to zero, at $P_x=1/2$, when both response distributions are identical and the response is independent of the stimulus. If error is conventionally restricted to $0\leq P_x\leq1/2$, the maximum occurs only at zero.
The symmetric-error assumption matters. A single overall error probability would not determine this information if the two stimuli had different conditional error rates or if the response included additional outcomes.
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