Predict first: If we multiply both w and b by 5, does the decision boundary move? What happens to the predicted probabilities?
1-D logistic model
Decision boundary x*—
P(Y=1 | query)—
Log-odds at query—
Predicted class—
Same boundary, different confidence
Reveal the key idea
The decision boundary is where wx+b=0. Multiplying both by a positive constant leaves that equation unchanged.
But the sigmoid sees a larger-magnitude logit away from the boundary, so probabilities become more extreme. The same 0/1 predictions can therefore have very different confidence.