Maths IA · Statistics and probability · topic 19 of 40
Screening tests: why a positive can still be wrong · IB Maths IA idea: guide, sample graph and step-by-step plan
gentle start, a real WOW momentA test that is right 90% of the time says you have a rare condition. Are you probably ill? Your first instinct will be wrong, and showing why is the whole IA.
- ✓ you like counter-intuitive results
- ✓ you enjoy tree diagrams
- ✓ you want a gentle first project
The big idea
the base rate is the secretWhen a condition is rare, most positive results come from the healthy majority, even for an accurate test. Bayes' theorem and tree diagrams quantify this. You take real published figures for prevalence, sensitivity and specificity, work out the chance a positive result is right, and show how it changes as prevalence, accuracy or a second test changes. The surprise is the point, but it needs careful justification: every number must have a source and every step a reason. The depth comes from sensitivity analysis, repeat testing, or a proof of the general result. At HL, build something beyond a single application of the formula.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Calculate the chance a positive is correct for a range of prevalences, from very rare to common.
→ Your graph shows the curve, and the conclusion names when a test becomes useful.Work out the chance of disease after two positive tests, using the first result as new information.
→ You use Bayes updating twice and discuss independence of the tests, which is a bigger assumption than it looks.Use published accuracy and prevalence from a named screening programme or a public health source.
→ Your numbers have sources, and you must discuss how reliable those figures are.Compare how much a change in each affects the result.
→ Your analysis ranks which test property matters most, a sensitivity analysis with a clear conclusion.Add costs for false positives and false negatives and compare tests by expected cost.
→ Expected value joins conditional probability, so your maths links two areas.Derive the posterior probability from the definition of conditional probability, then derive how it behaves for small prevalence.
→ A general argument gives the rigour and sophistication HL criterion E needs; a single worked example does not.Your Maths IA research question: run the check
Write your own question first. Then make sure it ticks every box:
- Names the condition or test, and says where its figures come from.
- Says which probability you want: given a positive result, what is the chance of the condition?
- Says what you will vary: prevalence, accuracy, number of tests.
- States what 'useful test' means in your question.
- Does not give medical advice; it is a maths question about probability.
The picture to draw
A tree diagram splits people by the condition, then by the test result. The question circles the two positive branches.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why this test; what people tend to assume; your prediction before calculating.
Conditional probability, the tree diagram, and Bayes' theorem with a justification of where it comes from.
Prevalence, sensitivity, specificity, each with a source and a comment on how reliable it is.
One complete calculation with a tree diagram, and the same answer in natural frequencies ('out of 10 000 people').
A table and a graph of the answer as prevalence (and other inputs) change; where the answer flips.
A second test, costs, or a general proof. At HL, this should carry the sophistication.
When a positive result should worry someone, in numbers and in plain language.
Independence of repeat tests, reliability of sources, and one next step with a reason.
Sources for every figure.
What your Maths IA graph could look like
Made-up test accuracy (90% sensitive), to show the form of a strong graph. example only
- The horizontal axis is prevalence, the one input people forget.
- Two curves for two specificities show which test property matters here.
- Sensitivity is fixed in this picture; say what you held constant.
- Mark your real source's prevalence with a vertical line, so the graph answers your question.
- Another graph to consider: the same quantity after a second positive test.
Where do the positive results come from?
natural frequencies help!Natural frequencies make the idea visible. Here 10 000 made-up people: 1% have the condition, the test catches 90% of them and wrongly flags 10% of the rest. Your numbers will come from a real source.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Choose a test and a sourceA screening test with published accuracy and a known prevalence. Keep every source.
- PredictWrite down your gut answer: if the test is positive, how likely is the condition? Keep it to compare later.
- Learn the mathsConditional probability, tree diagrams and Bayes' theorem. Do a toy example with coins or cards first.
- Trial runDo one full tree with round numbers (out of 10 000 people). Does it match your formula answer?
- Use real figuresInsert the real prevalence, sensitivity and specificity. Note the source and the year for each.
- Vary the inputsChange prevalence and each accuracy figure. Record how the answer moves, with a table and graph.
- Add a twistA second test, costs, or a general proof. Justify why you chose it.
- Check the logicDo tree probabilities sum to 1 at each split? Do your two methods agree? Does the answer make sense at extreme values?
- ReflectHow reliable are your sources? What if the tests are not independent? Reflect throughout.
- Write, then cutShow one tree and one calculation in full, then a table. Define each symbol, then check before your teacher's draft.
Maths you'll need
Traps to avoid in this Maths IA
the classic mix-up- Confusing P(positive | condition) with P(condition | positive).
- Figures with no source, or a source you cannot reach.
- Using a rare-condition example but only ever one prevalence.
- Assuming two repeat tests are independent without comment.
- Medical conclusions or advice; stay with the mathematics.
- At HL: one application of Bayes' formula; HL criterion E needs more depth, such as a proof, sensitivity analysis or links to expected value.
- A tree with branches that do not sum to 1.
More Maths IA ideas like this
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- Waiting times: does a Poisson model fit?●●○
- Polls: how big a sample shrinks the margin of error?●●○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.