Where the walk starts

Consider a hypothetical screening test. It detects 99% of people with a condition and gives a false positive for 1% of people without it. The condition affects 1 in 1,000 people. What is the probability that someone with a positive result has the condition?

Among 100,000 people, expect 99 true positives and 999 false positives. So the answer is $99/(99+999)\approx9\%$ . Most positive results come from the much larger group without the condition. Bayes’ rule expresses this calculation directly.

Evidence is a step. The prior is where you’re standing.

Chapter three established what a clue is worth: the test’s evidence is how much more often it says positive on the sick than on the healthy — it catches 99% of real cases and false-alarms on 1% of healthy people, so a positive multiplies your odds by 99. On the log ruler: +4.6 points. This is the evidence supplied by one positive result.

We also need the starting probability. Before the test, you weren’t at 50/50. The condition is 1-in-1,000 rare, so you started at odds of 1:999 — about −6.9 points, deep in “almost certainly not.” That starting position is the prior, which represents the information available before the result.

Bayes’ rule adds these quantities on the log-odds scale:

posterior log-odds = prior log-odds + log likelihood ratio
                   ≈ −6.9 + 4.6 = −2.3 → sigmoid → 9%

The positive result raises the probability from 0.1% to about 9%. That is a large increase, even though the final probability remains below 50%.

Believing is arithmetic

The posterior depends on both the starting probability and the strength of the evidence:

A weak prior and strong evidence → evidence wins. (Coin-flip prior + one positive test → 99%.)
A very small prior can leave a small posterior even after strong evidence. (1-in-100,000 + one positive → still only ~0.1%.)
Independent evidence, conditional on each hypothesis, contributes additional log likelihood ratios.

Go deeper: the multiplication you usually see

Textbooks write Bayes’ rule as a fraction: P(sick | positive) = P(positive | sick) · P(sick) / P(positive). It’s the same statement, still on the squashed ruler — which is exactly why it’s hard to feel. Rewrite it in odds and the fraction collapses into the form we walked:

posterior odds = prior odds × likelihood ratio

Taking logarithms turns the product into a sum: log(posterior odds) = log(prior odds) + log(likelihood ratio). These are equivalent forms of the same update.

Next: summarize a random quantity by its probability-weighted average, the expectation.

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