 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: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 entropywith $0\log 0=0$. Thus $H_{\mathrm{noise}}=H(R\mid S)=h_2(P_x)$, and the mutual information isIt 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 $0\le P_x\le 1/2$, 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.