Bayes' Theorem in Practice: Three Problems That Build the Intuition

Bayes' theorem: P(A|B) = P(B|A) · P(A) / P(B)

In words: the probability of hypothesis A given evidence B equals the probability of observing B if A were true, times the prior probability of A, divided by the total probability of observing B.

The formula is correct. Its significance is that it quantifies how evidence changes beliefs — and how much the prior probability matters.

Problem 1 — Medical testing

A disease affects 1% of the population. A test for the disease is 95% accurate (if you have the disease, the test is positive 95% of the time; if you do not have it, the test is negative 95% of the time).

You test positive. What is the probability you have the disease?

Most people say "95%" — because the test is 95% accurate. The correct answer is approximately 16%.

Using Bayes': P(disease | positive) = P(positive | disease) · P(disease) / P(positive)

P(positive) = P(positive | disease)·P(disease) + P(positive | no disease)·P(no disease) = 0.95 × 0.01 + 0.05 × 0.99 = 0.0095 + 0.0495 = 0.059

P(disease | positive) = (0.95 × 0.01) / 0.059 ≈ 0.161 ≈ 16%

The low prior probability (1% prevalence) dominates the calculation. This is the base-rate fallacy: intuition ignores the prior and focuses on the evidence alone.

Problem 2 — Card game

You draw two cards from a standard 52-card deck without replacement. You know (from a peek by someone else) that at least one card is an Ace. What is the probability that both are Aces?

P(both Aces | at least one Ace) = P(both Aces) / P(at least one Ace)

P(both Aces) = C(4,2)/C(52,2) = 6/1326 = 1/221

P(at least one Ace) = 1 − P(no Aces) = 1 − C(48,2)/C(52,2) = 1 − 1128/1326 = 198/1326 = 33/221

P(both | at least one) = (1/221) / (33/221) = 1/33 ≈ 3%

Problem 3 — Spam filtering

A spam filter classifies emails. 30% of incoming email is spam. The filter correctly identifies spam 90% of the time (true positive rate). It incorrectly marks legitimate emails as spam 2% of the time (false positive rate).

An email is marked spam. What is the probability it actually is spam?

P(spam | marked spam) = P(marked spam | spam) · P(spam) / P(marked spam) = (0.90 × 0.30) / (0.90 × 0.30 + 0.02 × 0.70) = 0.27 / (0.27 + 0.014) = 0.27 / 0.284 ≈ 95%

The much higher prior (30% spam rate vs. 1% disease rate) produces a much more reliable filter output. This is why spam filters are accurate and why disease tests need confirmatory tests.