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.
Properties of Probability Laws
If \(A \subset B\), then \(\mathbb{P}(A) \le \mathbb{P}(B)\).
$$ \mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B) $$
$$ \mathbb{P}(A \cup B) \le \mathbb{P}(A) + \mathbb{P}(B) $$
$$ \mathbb{P}(A \cup B \cup C) = \mathbb{P}(A) + \mathbb{P}(A^c \cap B) + \mathbb{P}(A^c \cap B^c \cap C) $$
$$ \mathbb{P}(A^c) = 1 - \mathbb{P}(A) $$
Axioms That Each Probability Law Obeys
Non-negativity. For every event \(A\), \(0 \le \mathbb{P}(A) \le 1 \)
Additivity. For disjoint events, \(\mathbb{P}(A_1 \cup A_2 \cup … ) = \mathbb{P}(A) + \mathbb{P}(A_2) + … \)
Normalization. \(\mathbb{P}(\Omega) = 1\)
Frequentist Approach
Keep repeating the experiment, and then measure the frequency of \(A\). For example, in the 2024 GSS survey, of the 3,986 respondents, 42 were bankers. If we choose a person from this population at random, the probability that they are a bank teller is \(42 / 3986 \approx 1.054\%\).
from __future__ import annotations
import pandas as pd
def probability(A: pd.Series[bool]) -> float:
"""Computes the probability of a proposition, A."""
return A.mean()
References
- ORF 309: Probability and Stochastic Systems. Mykhaylo Shkolnikov. 2017.
- 1. Probability — Think Bayes. Allen Downey. allendowney.github.io . Accessed Sep 17, 2026.