Conditional Probability & Bayes' Theorem

Calculate posterior probability using Bayes' Theorem: P(A|B) from prior, likelihood, and false positive rate.

The initial probability of event A before observing evidence B. Example: disease prevalence = 0.01 (1%).
Probability of observing B given A is true. Example: test sensitivity = 0.95 (95%).
Probability of observing B given A is false. Example: false positive rate = 0.05 (5%).

Bayesian Analysis

P(A) — Prior
P(¬A) — Prior Complement
P(B|A) — Likelihood
P(B|¬A) — False Positive
P(B) — Total Evidence
P(A|B) — Posterior
P(¬A|B) — Posterior Complement
Likelihood Ratio

Bayes' Theorem Formula

P(A|B) = P(B|A) × P(A) / P(B)
P(B) = P(B|A)×P(A) + P(B|¬A)×P(¬A)
References: Bayes' Theorem (1763): P(A|B) = P(B|A)P(A)/P(B). Law of total probability: P(B) = ∑P(B|A_i)P(A_i). Likelihood ratio = P(B|A)/P(B|¬A).