Conditional Probability

Definition

$ \mathbb{P}(A|B) $ is the probability of event $A$ given that event $B$ occurs:

$$ \mathbb{P}(A|B) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} $$

Notice that if $P(B) = 0$ then it’s meaningless to condition $A$ on $B$.

from __future__ import annotations

import pandas as pd

from .probability_defn import probability


def conditional_probability(
    proposition: pd.Series[bool], given: pd.Series[bool]
) -> float:
    # proposition[given] contains the values of proposition for given.
    return probability(proposition[given])

Satisfying the Axioms of Probability

Axiom 1: Normalization

$$ \mathbb{P}(B|B) = \frac{\mathbb{P}(B \cap B)}{\mathbb{P}(B)} = 1 $$

Axiom 2: Non-negativity

Because $A \cap B \subseteq B $, then $ 0 \le \mathbb{P}(A|B) \le 1 $.

Axiom 3: Additivity

Let $A$ and $C$ be mutually exclusive, then:

$$ (A \cup C) \cap B = (A \cap B) \cup (C \cap B) $$

… and $\mathbb{P}((A \cup C) | B)$ is:

$$ = \frac{ \mathbb{P}((A \cup C) \cap B) }{ \mathbb{P}(B) } $$ $$ = \frac{ \mathbb{P}(A \cap B) }{\mathbb{P}(B)} + \frac{ \mathbb{P}(C \cap B) }{\mathbb{P}(B)} $$ $$ = \mathbb{P}(A | B) + \mathbb{P}(C | B) $$

References

ORF 309: Probability and Stochastic Systems. Mykhaylo Shkolnikov. 2017.
1. Probability — Think Bayes. Allen Downey. allendowney.github.io . Accessed Sep 17, 2026.