Shkolnikov, Mykhaylo

Ingredients of Probability

There is an outcome space, \(\Omega\), e.g., the number of students in the first class: \( \Omega = \{ 0, 1, 2, 3, … \} \).

There are events of interest, e.g., at least one student shows up: \(A = \{1, 2, 3, … \} \).

A probability measure, \(\mathbb{P}\), takes events \(A\) and assigns them \(\mathbb{P}(A) \in [0, 1]\), such that the axioms of probability are satisfied.

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

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