Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 218 / 3 / b / Solution

The sigmoid function is strictly increasing. Therefore the risky region is $\hat{\eta }(x)\ge \log \{p/(1-p)\}$. Its decision boundary is the hyperplanePoints on or above this score threshold are assigned to the risky class. For $p=1/2$, the right side is zero.