Logistic regression — probability from a score

Background

Logistic regression turns a linear score (the logit) into a probability with the sigmoid:

$$z = \mathbf{w}^\top \mathbf{x} + b, \qquad \mathbb{P}(y=1 \mid \mathbf{x}) = \sigma(z) = \frac{1}{1 + e^{-z}}$$

The linear part is the same $\mathbf{w}^\top\mathbf{x}+b$ you met in linear regression; the sigmoid is the only new piece, and it guarantees the output lands in $(0, 1)$ so it can be read as a probability.

Problem statement

Implement predict_proba(x, w, b) that returns $\sigma(\mathbf{w}^\top\mathbf{x} + b)$ for a single sample.

Input

• x — array-like feature vector of length $n$.
• w — array-like weight vector of length $n$.
• b — float, the bias.

Output

Returns a Python float in $(0, 1)$: the predicted probability of class 1.

Examples

Example 1 — a zero score is a coin flip

Input:  x = [0, 0], w = [1, 1], b = 0
Output: 0.5

Explanation: $z = 0$, and $\sigma(0) = 0.5$ — exactly the toss-up point.

Example 2 — a positive score leans toward class 1

Input:  x = [1, 1], w = [1, 2], b = -1
Output: 0.8807970779...

Explanation: $z = 1\cdot1 + 2\cdot1 - 1 = 2$, and $\sigma(2) \approx 0.8808$.

Constraints

• Compute the dot product $\mathbf{w}^\top\mathbf{x}$, add b, then apply the sigmoid.
• Return a plain float.

Notes

• This is the entire logistic-regression forward pass for one example. Stacking many samples just turns the dot product into a matrix multiply $X\mathbf{w} + b$.