Bayes' theorem and applications: Worked examples

This page is a case zoo. The parent note gave you the formula and three examples. Here we deliberately hunt down every kind of Bayes problem you can meet — including the weird degenerate ones — and work each to the end.

Before symbols fly, one reminder of what each letter means, so you never have to scroll back:

Everything here rests on 1.3.01-Conditional-probability-and-independence and 1.3.02-Law-of-total-probability.

The scenario matrix

Every Bayes problem lands in one of these cells. The examples that follow are labelled by cell, and together they touch all of them.

Cell	What makes it distinct	Covered by
A. Rare-cause / low prior	prior tiny, test good → posterior still low	Ex 1
B. Balanced prior	prior near $0.5$ , evidence does the work	Ex 2
C. Sequential update	apply Bayes twice, posterior becomes new prior	Ex 3
D. Degenerate: perfect evidence	$P (E ∣ \neg H) = 0$ → posterior locks to $1$	Ex 4
E. Degenerate: useless evidence	$P (E ∣ H) = P (E ∣ \neg H)$ → belief unchanged	Ex 4
F. Boundary priors	$P (H) = 0$ or $P (H) = 1$ → belief frozen	Ex 5
G. Multiple hypotheses ( $n > 2$ )	normalize over a whole list	Ex 6
H. Naive-Bayes / product of features	many pieces of evidence multiply	Ex 7
I. Real-world word twist	you must extract the numbers first	Ex 8
J. Exam twist: reverse a false positive	given the posterior, solve for a hidden rate	Ex 9

The three curves you should keep in your head — how the posterior grows as evidence accumulates for different priors — are drawn here:

Figure — Bayes' theorem and applications

Notice the red curve (rare cause): even after strong evidence it climbs slowly. That is Cell A in one picture.

Worked examples

Cell A — Rare cause, good test

Worked example Ex 1 · The classic disease test, re-examined

A disease affects $0.5%$ of people. A test has sensitivity $P (T ∣ D) = 0.99$ and false-positive rate $P (T ∣ \neg D) = 0.02$ . You test positive. Probability you have it?

Forecast: the test is 99% accurate — guess the answer before reading. Most people say "about 99%". Write your guess down.

Step 1. List the pieces. $P (D) = 0.005, P (\neg D) = 0.995, P (T ∣ D) = 0.99, P (T ∣ \neg D) = 0.02.$ Why this step? Bayes needs exactly these four numbers; naming them prevents mixing up $P (T ∣ D)$ with $P (D ∣ T)$ .

Step 2. Numerator = "true positives" weight. $P (T ∣ D) P (D) = 0.99 \times 0.005 = 0.00495.$ Why this step? This is the slice of people who are sick and test positive.

Step 3. Other term = "false positives" weight. $P (T ∣ \neg D) P (\neg D) = 0.02 \times 0.995 = 0.0199.$ Why this step? Healthy people who nonetheless test positive. They dilute the good news.

Step 4. Divide. $P (D ∣ T) = \frac{0.00495}{0.00495 + 0.0199} = \frac{0.00495}{0.02485} \approx 0.1992.$

Answer: ≈ 19.9%. Far below your gut guess.

Verify: false positives ( $0.0199$ ) outnumber true positives ( $0.00495$ ) roughly $4 : 1$ , so the posterior should sit near $\frac{1}{5}$ — and $0.199 \approx \frac{1}{5}$ . ✓ Consistent with the red curve in the figure. See 2.1.05-Naive-Bayes-classifier for the same maths at scale.

Cell B — Balanced prior, evidence decides

Worked example Ex 2 · Two urns

Urn $A$ holds 3 red, 1 blue. Urn $B$ holds 1 red, 3 blue. You pick an urn at random ( $P (A) = P (B) = 0.5$ ) and draw one ball: it is red. Which urn?

Forecast: guess the probability it was urn $A$ .

Step 1. Priors and likelihoods. $P (A) = P (B) = 0.5, P (red ∣ A) = \frac{3}{4}, P (red ∣ B) = \frac{1}{4} .$ Why this step? " $3$ of $4$ balls red" is the likelihood — it's a direct count.

Step 2. Numerator. $P (red ∣ A) P (A) = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} .$

Step 3. Evidence (denominator) via law of total probability. $P (red) = \frac{3}{4} \cdot \frac{1}{2} + \frac{1}{4} \cdot \frac{1}{2} = \frac{3}{8} + \frac{1}{8} = \frac{1}{2} .$ Why this step? Red can come from either urn; sum both paths.

Step 4. Divide. $P (A ∣ red) = \frac{3/8}{1/2} = \frac{3}{4} = 0.75.$

Answer: 75%. With equal priors the update ratio equals the likelihood ratio $3 : 1$ .

Verify: by symmetry $P (B ∣ red) = \frac{1}{4}$ , and $0.75 + 0.25 = 1$ . ✓ Posteriors sum to $1$ .

Cell C — Sequential updating

Worked example Ex 3 · Two independent tests

Same disease as Ex 1 ( $P (D) = 0.005$ ). You take the test twice, independently, and both are positive. Now what?

Forecast: should two positives push you above or below 50%? Guess.

Step 1. First update — this is exactly Ex 1: posterior $= 0.1992$ . Why this step? Sequential Bayes reuses old work; the first posterior becomes the new prior.

Step 2. New prior $P (D) = 0.1992, P (\neg D) = 0.8008$ . Same likelihoods. Why this step? Tests are independent given true status, so the second test is scored the same way.

Step 3. Second update. $P (D ∣ T_{2}) = \frac{0.99 \times 0.1992}{0.99 \times 0.1992 + 0.02 \times 0.8008} .$ Numerator $= 0.19721$ , other term $= 0.016016$ . $= \frac{0.19721}{0.21322} \approx 0.9249.$

Answer: ≈ 92.5%. Two independent positives overwhelm the low prior.

Verify (shortcut): the odds form $odds_{post} = odds_{prior} \times LR^{2}$ where $LR = 0.99/0.02 = 49.5$ . Prior odds $= 0.005/0.995$ . So post-odds $= \frac{0.005}{0.995} \times 49. 5^{2} = 12.318$ , giving $P = \frac{12.318}{13.318} = 0.9249$ . ✓ Matches. This is the incremental spirit of 4.2.03-Bayesian-inference.

Cells D & E — Degenerate evidence

The way the likelihood ratio stretches the prior is drawn here:

Watch: ratio $= 1$ (mint dot) sits on the diagonal — no change. Large ratios bend the curve up; ratios below $1$ bend it down.

Cell F — Boundary priors freeze belief

Cell G — Many hypotheses

Worked example Ex 6 · Three factories

Bulbs come from three factories: $A$ makes $50%$ , $B$ makes $30%$ , $C$ makes $20%$ of output. Defect rates: $A : 2%$ , $B : 3%$ , $C : 5%$ . A bulb is defective. Which factory most likely made it?

Forecast: biggest producer $A$ ? Or dirtiest producer $C$ ?

Step 1. Joint weights $P (def ∣ F) P (F)$ : $A : 0.02 \times 0.5 = 0.010, B : 0.03 \times 0.3 = 0.009, C : 0.05 \times 0.2 = 0.010.$ Why this step? Extended Bayes: each factory is one hypothesis; compute its numerator.

Step 2. Evidence = sum $= 0.010 + 0.009 + 0.010 = 0.029$ . Why? Normalizer so the three posteriors sum to $1$ .

Step 3. Posteriors: $P (A ∣ def) = \frac{0.010}{0.029} \approx 0.345, P (B) = \frac{0.009}{0.029} \approx 0.310, P (C) = \frac{0.010}{0.029} \approx 0.345.$

Answer: $A$ and $C$ tie at $34.5%$ — the big-clean factory and the small-dirty factory are equally suspect.

Verify: $0.345 + 0.310 + 0.345 = 1.000$ . ✓ Sums to one.

Cell H — Naive Bayes product

Cell I — Real-world word twist

Worked example Ex 8 · The taxicab problem

A city has $85%$ Green cabs and $15%$ Blue cabs. A witness identifies a cab as Blue. In tests the witness is correct $80%$ of the time (so wrong $20%$ ). What is the probability the cab really was Blue?

Forecast: witness is 80% reliable — so 80%? (This is the classic trap.)

Step 1. Extract the hidden Bayes structure. Hypothesis $B$ = "cab is Blue". Evidence $W$ = "witness says Blue". $P (B) = 0.15, P (\neg B) = 0.85.$ $P (W ∣ B) = 0.8$ (correct on a Blue cab). $P (W ∣ \neg B) = 0.2$ (calls a Green cab Blue by error). Why this step? The percentages are reliabilities, not posteriors — you must map them to likelihoods first.

Step 2. Numerator $= 0.8 \times 0.15 = 0.12$ . Rival $= 0.2 \times 0.85 = 0.17$ .

Step 3. $P (B ∣ W) = \frac{0.12}{0.12 + 0.17} = \frac{0.12}{0.29} \approx 0.4138.$

Answer: ≈ 41.4% — more likely Green than Blue! Base rate of Green cabs beats the witness.

Verify: $P (G ∣ W) = \frac{0.17}{0.29} = 0.5862$ and $0.4138 + 0.5862 = 1$ . ✓ The base-rate neglect trap in action.

Cell J — Exam twist: solve for a hidden rate

Worked example Ex 9 · Reverse-engineer the false-positive rate

A screening test has sensitivity $P (T ∣ D) = 0.9$ on a disease with prior $P (D) = 0.1$ . The lab reports that a positive test gives posterior $P (D ∣ T) = 0.5$ . What false-positive rate $f = P (T ∣ \neg D)$ produces this?

Forecast: guess whether $f$ is bigger or smaller than the sensitivity.

Step 1. Write Bayes with $f$ unknown. $0.5 = \frac{0.9 \times 0.1}{0.9 \times 0.1 + f \times 0.9} = \frac{0.09}{0.09 + 0.9 f} .$ Why this step? We invert the usual problem: the posterior is given, the rate is the unknown.

Step 2. Cross-multiply: $0.5 (0.09 + 0.9 f) = 0.09$ , so $0.045 + 0.45 f = 0.09$ .

Step 3. Solve: $0.45 f = 0.045 \Rightarrow f = 0.1.$

Answer: false-positive rate $= 0.1$ (10%).

Verify (plug back): $\frac{0.9 \times 0.1}{0.9 \times 0.1 + 0.1 \times 0.9} = \frac{0.09}{0.18} = 0.5$ . ✓ Recovers the given posterior. This algebraic inversion is cousin to 3.4.01-Maximum-likelihood-estimation where we solve for unknown parameters.

Quick self-test

Recall Which cell? (reveal to check)

A test never fires on healthy people — what is your posterior after a positive? ::: $1$ (Cell D, perfect discriminator). Prior is $0.5$ and likelihood ratio is $1$ — new belief? ::: Still $0.5$ (Cell E, useless evidence). You start certain, $P (H) = 1$ — can evidence move you? ::: No, belief is frozen (Cell F). Two independent positives on a rare disease pushed $0.199 \to$ ? ::: $\approx 0.925$ (Cell C).

Connections

Bayes' theorem and applications — the parent formula and theory
1.3.01-Conditional-probability-and-independence — the definition every example rests on
1.3.02-Law-of-total-probability — builds every denominator here
2.1.05-Naive-Bayes-classifier — Cell H at industrial scale
4.2.03-Bayesian-inference — sequential updating (Cell C) as a worldview
3.4.01-Maximum-likelihood-estimation — the reverse-solve flavour of Cell J
1.3.04-Random-variables-and-distributions — extends these cells to continuous data