Rearrange the logistic function [Pleft(T_{text {new }}=1 mid mathbf{x}_{text {new }}, widehat{mathbf{w}} ight)=frac{1}{1+exp left(-widehat{mathbf{w}}^{top} mathbf{x}_{text {new }}

Rearrange the logistic function

\[P\left(T_{\text {new }}=1 \mid \mathbf{x}_{\text {new }}, \widehat{\mathbf{w}}\right)=\frac{1}{1+\exp \left(-\widehat{\mathbf{w}}^{\top} \mathbf{x}_{\text {new }}\right)}\]

to show that \(P\left(T_{\text {new }}=1 \mid \mathbf{x}_{\text {new }}, \widehat{\mathbf{w}}\right)=0.5\) implies \(\widehat{\mathbf{w}}^{\top} \mathbf{x}_{\text {new }}=0\).