Sigmoid: what is p when z = 0?

Sigmoid maps a linear score $z$ to a probability $p \in (0, 1)$:

At $z = 0$: $e^{0} = 1$, so $p = 0.5$. The neuron is maximally undecided — neither spam nor normal. Beginners often think $z = 0$ means “definitely not spam”; it is the midpoint.

Numeric examples: $z = 2.0 \Rightarrow p \approx 0.88$; $z = -2.0 \Rightarrow p \approx 0.12$. The curve saturates: large $|z|$ pushes $p$ near 0 or 1 but never hits the boundary — useful for training gradients.

import math
def sigmoid(z):
    return 1 / (1 + math.exp(-z))
assert abs(sigmoid(0) - 0.5) < 1e-9

When filtering mail you compare $p$ to a threshold (say $0.5$). Knowing $z = 0 \Rightarrow p = 0.5$ explains the quiz without memorizing answers. If you raise the threshold to $0.7$, you label spam only when $z$ is large enough for sigmoid to pass 0.7 — a business choice, not a property of the curve. Lesson two in the course reuses the same formula on the toy inbox vectors, so the quiz question about $z=0$ is tied to real feature counts, not a trick number.