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Bayes' Theorem — Positive Predictive Value

Answer: just 16.7%. Even doctors get this question wrong half the time — and the deciding factor isn't sensitivity or specificity, but **prevalence**. Trace the numbers in the "town of 1,000" diagram below.

P.04 / BAYES THEOREM

Bayes' Theorem — The Posterior Plot Twist

Distribution toolbox complete. Finally, Bayes' theorem flips the conditioning. You test positive — what's the chance you're actually sick? Intuition fails here; let's build it.

"The test has 99% sensitivity & 95% specificity, and you tested positive" — is there a 99% chance you're sick?
…Answer: only 16.7%. More than half of doctors get this classic quiz wrong.
Walk through the numbers. In a town of 1,000, 10 people are sick and 990 are healthy. Test everyone and 60 come back positive — 10 truly sick (true positives) + 50 healthy but wrongly flagged (false positives). If you're one of those 60, your chance of actually being sick is 10 ÷ 60 = 16.7%. The larger the healthy majority, the more false positives dilute the real cases.

Experiment Guide — try these in order
  1. Step 1: Sweep prevalence between 0.001 (general population) and 0.4 (high-risk group) → with the same 99% sensitivity and 95% specificity, PPV swings from 1.96% to 92.86%. Feel how strongly the prior dominates.
  2. Step 2: At prevalence 0.001, PPV is just 1.96% — only ~2 of 100 positives are actually sick. Feel the gap from intuition.
  3. Step 3: Keep prevalence at 0.001 but raise specificity to 99.9% → false positives plummet, PPV improves dramatically.
  4. Step 4: Watch the "town of 1,000" diagram and compare TP (true positive) vs FP (false positive) counts.
True positives (sick & tested +)—
False positives (healthy but tested +)—
If you tested +, chance you're sick—
If you tested NEG, chance you're healthy—

// Formula used here

Here A = "truly sick" and B = "tested positive"
P(A|B) = probability of being sick given a positive test (PPV), P(B|A) = sensitivity, P(A) = prevalence, P(B) = overall probability of testing positive.
When prevalence is tiny the numerator shrinks while the denominator, swollen with false positives, stays large — that's exactly the "10 out of 60" above.

// Common misconceptions

❌ "Positive test = disease"

With 1% prevalence, 95% sensitivity, and 90% specificity, the positive predictive value is only about 8.8%. Of 100 people who test positive, roughly 9 actually have the disease — the other 91 are healthy false positives. Drag the prevalence slider down in the simulation above and watch this ratio shift before your eyes.

❌ "High sensitivity and specificity are enough"

Screen a low-prevalence population and even with 99% sensitivity and 99% specificity, the positive predictive value is only about 50%. Who you test (the prior probability) matters as much as how good the test is.

❌ "Bayes' theorem is a one-shot calculation"

The posterior from the first test can become the prior for the second. Two consecutive positives dramatically raise the posterior probability. This is the idea behind Bayesian updating.

// Shapes you'll meet again

Bayes' theorem keeps reappearing in the same handful of shapes.

  • The classic scenario shape: "Prevalence p%, sensitivity a%, specificity b%. Given a positive test, what's the probability of disease?" — the numbers change, the structure doesn't
  • Total probability sitting in the denominator: is the shape Bayes' denominator always takes. The "true positives + false positives" picture above is this same equation
  • The natural-frequency picture: split 1,000 people into a table and the same problem looks like simple counting instead of fractions

The law of total probability that assembles Bayes' denominator P(B) lives at Probability Rules.

// Related topics

  • Probability Rules — conditional probability and the law of total probability assemble Bayes' denominator and bridge it to the posterior
  • Hypothesis Testing — false positives like "positive yet not diseased" line up in the same probability space as Type I error α
  • Discrete & Exponential — putting sensitivity, specificity, and prevalence onto a probability model pulls binomial and Poisson out from underneath
  • Confidence Interval — sensitivity, specificity, and prevalence are estimated from samples — CIs hold the uncertainty in interval form
  • "Type I vs Type II" column — lays the error 2×2 and PPV onto a single page so that false positives and Type I error overlap in the same probability space

// Further reading

Type I vs Type II Errors — One 2×2 Table Sorts Them Out α, β, power, and effect size all live on a single 2×2 picture — confusion disappears

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