Enrichment · STEM Explorations

A positive test
is not a yes.

Set how common something is and how often a clue turns up when it is true and when it is not. Guess the new chance, then check yourself against a count of 1,000 cases, a tree and the arithmetic. Add more clues and watch the chance move.

Open the visualizerWhat you get

The visualizer

The worked example is a made-up screening test for a condition that 1 person in 100 has. Lock in a guess before you look at the answer. Five more examples are in the menu.

Loading the visualizer. If nothing appears, turn on JavaScript.

What you get

Enrichment · STEM ExplorationsFree

What you finish with
The chance that something is true after one clue or a chain of clues, with the working shown three ways (a count of cases, a frequency tree and the formula), and a record of how far your own first guess was from it.
Who it is for
High school and college students in statistics, probability, biology or psychology; math and science clubs; anyone trying to make sense of a test result or a forecast; teachers who want a live demonstration of base rates.
Format
Browser tool with three parts: one clue, a chain of up to eight clues, and saved setups. Prints the working; downloads the chain as a spreadsheet (.csv) and a file you can import on another device.
Access
Free. No account or sign-in. It runs in your web browser on a phone, tablet or computer. Everything you type stays in this browser and is saved there as you go; nothing is sent to Caveat.
Where it lives
The STEM Explorations space in Caveat Enrichment.
Support
Questions, a correction or a request: team@caveatacademic.com.

How to use it

  1. Read the worked example and lock in a guess before you look.
  2. Compare your guess with the answer, then turn on Show only the positive test to see who is left.
  3. Move the three numbers. Watch what a rarer condition or a test with fewer false alarms does to the answer.
  4. Open A chain of clues to add evidence one piece at a time and see the chance after each.
  5. Use Start blank for your own problem, change the words to match it, and save or print the result.

Open the visualizer

How the update works

One division, shown three ways.

The rule

Bayes’ rule answers one question: after a clue shows up, what share of all the ways that clue can show up are cases where the thing is true. Written with probabilities, P(true | clue) = P(true) × P(clue | true) ÷ [P(true) × P(clue | true) + P(not true) × P(clue | not true)].

In the worked example that is 1% × 90% ÷ (1% × 90% + 99% × 9%) = 0.9% ÷ 9.81% = 9.2%. The squares and the tree show the same division with whole people: 9 of 98.

Odds make chains short

The strength of a clue is its likelihood ratio: how many times as likely the clue is when the thing is true as when it is not. Multiply the odds you started with by that ratio and you have the new odds. A second clue multiplies again, which is why the order does not matter. The chain assumes each clue is independent of the others once the truth is known. Real clues often overlap, such as a repeat of the same test on the same sample, and then they count for less.

Words you may see elsewhere

The starting chance is the prior or base rate, and the result is the posterior. For a medical test, the chance of a positive result when the condition is present is the test’s sensitivity, and the chance of a positive when it is absent is 1 minus its specificity. The share of positives that are true is the positive predictive value.

How it is checked

The squares use whole-number arithmetic, so the four groups always add up to the total. Before each release the calculator is run against exact fractions on 4,500 random cases and has to agree on every one.

Two ways to get a positive test9 have the condition89 do not have itall 98 positive tests9 ÷ 98 = 9.2%chance of the condition
Why a rare thing stays unlikely. In the worked example the test is right 90% of the time for people who have the condition. But those people are rare, so the small share of false alarms among everyone else adds up to far more positive tests. Count both routes, then divide.

Sources

  1. Grinstead, C. M., & Snell, J. L. (1997). Introduction to Probability (2nd rev. ed.). American Mathematical Society. Chapter 4, Conditional Probability: Bayes’ formula and Bayes probabilities.
  2. Illowsky, B., & Dean, S. (2023). Introductory Statistics 2e. OpenStax, Rice University. Chapter 3, Probability Topics: conditional probability, contingency tables and tree diagrams.
  3. Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102(4), 684–704. Why counts of cases are easier to reason with than percentages.
  4. Tversky, A., & Kahneman, D. (1982). Evidential impact of base rates. In D. Kahneman, P. Slovic & A. Tversky (Eds.), Judgment Under Uncertainty: Heuristics and Biases (pp. 153–160). Cambridge University Press. The cab problem.

Every example uses made-up numbers. This page does arithmetic; it does not interpret a real medical test or any other real result. For the accuracy of a real test, and for what a result means for you, ask the clinician or the laboratory that ordered it. Support: team@caveatacademic.com.