Statistics, data and evidence · Methods and tools

Should new evidence change what we believe according to a fixed mathematical rule?

Bayes' theorem says exactly how a rational person should update a belief when new evidence arrives. Applied to medical tests, it gives results many people find surprising. This question asks whether rational belief can be reduced to a formula.

Claims

Bayes' theorem is a consequence of the rules of probability, so ignoring it means reasoning inconsistently.
It explains common mistakes, such as ignoring how rare a condition is, and so improves real decisions.

Counterclaims

The formula needs prior probabilities, which are often guesses, so its output can be no better than the judgement that went in.
People's beliefs also depend on trust, values and context that are hard to put into numbers.

Real-life situations from mathematics

Screening tests

If a disease affects 1 in 1,000 people and a test detects it 99% of the time with a 1% false-positive rate, most people who test positive do not have the disease. Studies have found that many doctors misjudge this kind of question.

Spam filters

Early email spam filters used Bayes' theorem to update the probability that a message is spam from the words it contains, an everyday example of rule-based updating.

Check dates and figures in a reliable source before you use them, and cite that source.

Use this in your TOK work

Essay. Fits titles about evidence, belief and changing our minds.

Exhibition. A home test kit's leaflet with accuracy figures is a strong object for a prompt about how we know what we know.

Link it to the prescribed title or the exhibition prompt you are working on, in your own words. See using maths examples in your essay and choosing exhibition objects.

Themes and study heading

Knowledge and technology Knowledge and the knower Methods and tools

How mathematical knowledge is produced and justified.