The instinct says: very likely, maybe 90%. Research on how people handle numbers like these, doctors included, finds that most get them badly wrong. If the condition affects 1 in 100 people of your age, the real answer is about 9%.
That gap, between what the evidence feels like it says and what it actually says, is what Bayesian thinking fixes. It isn't a formula you have to memorize. It's a habit: start from how likely something was before the new evidence, then ask how much the evidence should move you.
Doing it without any algebra
The trick, championed by the German psychologist Gerd Gigerenzer and colleagues, is to stop thinking in percentages and think about a crowd of real people. Their studies found that people, experts included, reason far more accurately when the same numbers are given as "natural frequencies," counts of people, rather than as probabilities.
Imagine 1,000 people taking that test:
| Have the condition | Don't have it | |
|---|---|---|
| People | 10 (1 in 100) | 990 |
| Test positive | 9 (90% of 10) | about 89 (9% of 990) |
Ninety-eight people test positive, and only 9 of them are actually ill. So a positive result means about a 9 in 98 chance, roughly 9%. The test is good. The problem is that the healthy group is so large that even a small error rate produces far more false alarms than there are true cases.
This is not an argument against testing. It's why a positive screening result usually leads to a second, more precise test rather than straight to treatment, and why the numbers matter when you're the one being tested. Gigerenzer's team gives another example: the claim that mammography screening "reduces the risk of dying from breast cancer by 25%" turns out, in their analysis, to mean about 1 fewer death per 1,000 women screened. Both statements can be true. Only one helps you decide.
The question is never just "how well does the evidence fit my idea?" It's "how well does it fit my idea compared with the alternatives, given how common each was to begin with?"
The two ingredients
Every Bayesian update has two parts:
1. The prior: how likely was this before the new evidence? Before the test, your chance was 1 in 100, because that's how common the condition is. Ignoring this starting point is called base-rate neglect, one of the biases covered in The Biases That Fool Everyone, this category.
2. The strength of the evidence: how much more likely is this evidence if the idea is true than if it's false? A positive result is 10 times more likely if you're ill (90%) than if you're healthy (9%). That ratio, called the likelihood ratio, is what tells you how far to move.
Put together, the rule is short: your new odds = your old odds × the strength of the evidence. Old odds of 1 to 99, times 10, gives about 10 to 99, roughly 9%. A second independent positive test multiplies by 10 again, and now you're at about 50–50. Evidence accumulates; you don't have to get it all from one clue.
Everyday uses
Reading someone's mood. A colleague sends a curt two-line email. Is she angry with you? Start with the prior: in an ordinary week, maybe there's a 1 in 10 chance she's annoyed with you. Then the evidence: curt emails might be twice as common when she's annoyed as when she's simply busy. Odds go from 1 to 9 to 2 to 9, about 18%. Worth noticing, not worth a sleepless night. If she also skips your meeting, update again.
Judging a news story. A dramatic claim surprises you. Before asking whether the evidence fits, ask how often claims like this, from sources like this, have turned out true. A strong prior against something isn't closed-mindedness; it's memory. (See How to Spot Bullshit, this category.)
Diagnosing a problem. Your car won't start on a cold morning. A dead battery and a failed starter both explain it, but dead batteries are far more common, especially in the cold. Check the common thing first.
Your own plans. The best guide to how long your project will take is usually how long similar projects took, yours and other people's, not your detailed plan for this one. Start there, then adjust for what's genuinely different.
Two rules worth keeping
Never be 100% sure. If your prior is exactly 0 or exactly 1, no evidence can ever move it: anything times zero is still zero. The statistician Dennis Lindley named this "Cromwell's rule," after Oliver Cromwell's 1650 plea to the Church of Scotland: "I beseech you, in the bowels of Christ, think it possible that you may be mistaken." Leave yourself a sliver of doubt about almost everything.
Extraordinary claims need extraordinary evidence. Carl Sagan made the phrase famous in the late 1970s, but the idea is much older. David Hume made the argument about miracles in 1748, and the mathematician Pierre-Simon Laplace wrote that the weight of evidence for an extraordinary claim must match its strangeness. In Bayesian terms: the lower the prior, the stronger the evidence has to be to get you to "probably true."
How the best forecasters do it
In a US government-funded forecasting tournament that ran from 2011, the Good Judgment Project, led by Philip Tetlock and Barbara Mellers, recruited ordinary volunteers to predict world events. The project beat the other research teams in the tournament by a wide margin, and its best volunteers, the "superforecasters," were the most accurate of all. A 2020 study of the project's data had a title that sums up one of the reasons: "Small steps to accuracy: Incremental belief updaters are better forecasters." They didn't swing wildly with every headline, and they didn't ignore news either. They moved a little, often.
The honest limits
Priors can be hard to know, and two reasonable people may start from different ones. Bayesian reasoning doesn't remove that disagreement; it makes it visible, which is usually more useful than pretending it isn't there. It also depends on honest estimates of evidence strength, and it's easy to fool yourself about those. And it's a habit of mind, not a demand to put numbers on everything. Rough numbers ("about 1 in 10," "twice as likely") are enough to stop most of the big mistakes.
The method is named after Thomas Bayes, an English minister who worked out the core idea. He never published it: his friend Richard Price found the essay after Bayes died in 1761 and presented it to the Royal Society in 1763. Laplace later developed it into the general approach we use today.
The takeaway
Bayesian thinking means starting from how likely something was before the evidence (the prior), then moving by how much more likely the evidence is if the idea is true than if it's false. Thinking in counts of people ("of 1,000 who take the test...") makes this easy and exposes how rare conditions plus decent tests still produce mostly false alarms. Never be completely certain, demand more evidence for more surprising claims, and update in small, frequent steps.
Try this: the next time something worries you, an odd symptom, a rumour, a curt message, write down two numbers before reacting: how likely this was before (out of 10 or 100), and how many times more likely the evidence is if the worry is true than if it isn't. Multiply, and see where you land.
Sources & Further Reading
Sources
- Natural frequencies: Hoffrage U., Lindsey S., Hertwig R. & Gigerenzer G., "Communicating statistical information," Science 290 (2000).
- Mammography "25%" equals about 1 in 1,000: Gigerenzer G. et al., "Helping doctors and patients make sense of health statistics," Psychological Science in the Public Interest 8 (2007).
- The worked test example (1% prevalence, 90% sensitivity, 9% false-positive rate) is illustrative; its numbers were computed directly.
- Cromwell's rule: Dennis Lindley; Cromwell's letter of 3 August 1650 to the General Assembly of the Church of Scotland.
- "Extraordinary claims": Sagan in Broca's Brain (1979) and Cosmos (1980); Hume, "Of Miracles" (1748); Laplace's essays (1810, 1814).
- Good Judgment Project and Atanasov P., Witkowski J., Ungar L., Mellers B. & Tetlock P., "Small steps to accuracy: Incremental belief updaters are better forecasters," Organizational Behavior and Human Decision Processes 160 (2020).
- Bayes's death (1761) and the essay's presentation by Richard Price (1763): standard histories of probability.
Go deeper
- 📘 Risk Savvy: How to Make Good Decisions — Gerd Gigerenzer (natural frequencies and medical numbers for non-mathematicians)
- 📘 Superforecasting — Philip Tetlock & Dan Gardner (how the best forecasters actually update)
- ▶️ 3Blue1Brown: "Bayes theorem, the geometry of changing beliefs" (a beautiful visual explanation)
- 🌐 See also: Probability: How to Not Fool Yourself (Mathematics category), for Monty Hall and the birthday paradox