Logistic regression — two-class probabilities

Background

In binary classification the model predicts $\mathbb{P}(y=1 \mid \mathbf{x})$. Because there are only two classes, the other probability is fixed by it:

$$\mathbb{P}(y=0 \mid \mathbf{x}) = 1 - \mathbb{P}(y=1 \mid \mathbf{x})$$

The two probabilities are non-negative and sum to 1 — exactly the $P=0.4$ / $P=0.6$ bars in the lesson. One number determines both.

Problem statement

Implement class_probabilities(p1) that, given $p_1 = \mathbb{P}(y=1 \mid \mathbf{x})$, returns the pair $(\mathbb{P}(y=0), \mathbb{P}(y=1))$.

Input

• p1 — float in $[0, 1]$, the probability of class 1.

Output

Returns a tuple of two Python floats (p0, p1) where p0 = 1 - p1. They sum to 1.

Examples

Example 1

Input:  p1 = 0.6
Output: (0.4, 0.6)

Explanation: the lesson's bars — class 1 at $0.6$ forces class 0 at $1-0.6=0.4$.

Example 2

Input:  p1 = 1.0
Output: (0.0, 1.0)

Explanation: full confidence in class 1 leaves zero for class 0.

Constraints

• Return plain Python floats in the order (p0, p1).
• Do not clamp or renormalise — assume p1 is already a valid probability in $[0, 1]$.

Notes

• This complement rule is why binary logistic regression needs only a single sigmoid output: predict $\mathbb{P}(y=1)$ and the rest follows. The multi-class generalisation that produces a whole vector summing to 1 is the softmax.