Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

A positive test result: how likely is it that you really have the condition?

AA HLAA SLAI SLAI HL Solid Also in: Probability

Research question

For a screening test with published sensitivity and specificity, how does the probability that a positive result is correct depend on how common the condition is, and how much does a second independent test help?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

Even an accurate test can give mostly false positives when a condition is rare — a result that surprises many doctors in surveys. Bayes' theorem makes this precise and has real consequences for screening policy.

The mathematics you'll need

Conditional probability and tree diagrams
Bayes' theorem (AA HL; explain for others)
Positive predictive value as a function of prevalence
Repeated testing and independence assumptions

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Chi-squared test for independence. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

Use published sensitivity, specificity and prevalence figures for a real screening programme (cite the health authority).

WHO Global Health Observatory — 1,000+ health indicators by country (immunisation, mortality, risk factors).

A possible outline

Explain sensitivity, specificity and prevalence.
Derive the positive predictive value.
Graph it against prevalence.
Model a second test.
Reflect on why tests are targeted at high-risk groups and on independence of repeat tests.

Pitfalls that cost marks

Confusing P(positive | ill) with P(ill | positive).
Unsourced test figures.
Assuming repeat tests are independent without comment.

Showing personal engagement

Survey adults' intuition with a scenario and compare with the truth.
Choose a test relevant to someone you know (without personal medical details).
Design a clear graphic for patients.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Conditional probability and tree diagrams; Bayes' theorem (AA HL; explain for others)
AA HL	Good fit	Conditional probability and tree diagrams; Bayes' theorem (AA HL; explain for others)
AI SL	Good fit	Conditional probability and tree diagrams; Bayes' theorem (AA HL; explain for others)
AI HL	Good fit	Conditional probability and tree diagrams; Bayes' theorem (AA HL; explain for others)

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: confusing P(positive | ill) with P(ill | positive); and unsourced test figures. Say how each one affects your answer.

Use of mathematics (E)

SL: conditional probability and tree diagrams; bayes' theorem (AA HL; explain for others); positive predictive value as a function of prevalence; repeated testing and independence assumptions, each explained in your own words and carried out correctly.

How health & biology explorations reach the top bands in general: Health, biology & medicine IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Add a cost for each outcome and find the prevalence at which screening everyone is worthwhile.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL health & biology exploration adds.

See a complete IA, marked

Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) is a complete health & biology exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA HL, AA SL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Conditional probability and tree diagrams, bayes' theorem (AA HL; explain for others), positive predictive value as a function of prevalence and repeated testing and independence assumptions.

Where can I get data for this IA?

Use published sensitivity, specificity and prevalence figures for a real screening programme (cite the health authority). Checked sources and tools: WHO Global Health Observatory.

Turn this idea into your IA

The planner opens with “A positive test result: how likely is it that you really have the condition?” filled in: the question, the maths and the data sources.

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