Module 3 — Conditional probability and evidence
EDU-MAT-201 · Module 3 of 4 · Estimated effort: 2 hours including practice
Before you begin: Complete Module 2 or demonstrate its evidence criteria. Use the course environment and downloads.
Outcomes
Calculate conditional probability using natural frequencies; distinguish sensitivity from positive predictive value; explain how prevalence changes an interpretation.
Explain and demonstrate
Conditional probability changes the group used as the denominator. In a fictional screening example, sensitivity asks what fraction of people with a condition test positive. The probability of having the condition after a positive result asks a different question: among all positive results, how many belong to people with the condition? Confusing these quantities creates confident but incorrect explanations.
Use a fictional population of 10,000, prevalence 1%, sensitivity 90% and false-positive rate 5%. Exactly 100 have the condition in this constructed example. Of those, 90 test positive. Of the other 9,900, 495 test positive. There are therefore 585 positive results, and 90 / 585 = 2 / 13, approximately 15.38%, are true positives. These are invented teaching values, not clinical advice or measured test characteristics.
Worked example: a frequency table
| Group | Positive | Negative | Total |
|---|---|---|---|
| Condition present | 90 | 10 | 100 |
| Condition absent | 495 | 9,405 | 9,900 |
| Total | 585 | 9,415 | 10,000 |
Ask learners to point to the denominator before entering numbers into a calculator. For sensitivity, select the “condition present” row. For probability after a positive result, select the “positive” column. The visual selection explains the difference more clearly than memorizing two formulas.
Changing prevalence while keeping sensitivity and false-positive rate fixed changes the positive-result interpretation. At 10% prevalence in the same population, true positives become 900 and false positives 450. The resulting probability is 900 / 1,350, or two thirds. The assumptions are deliberately held constant for comparison; real settings may also change the test characteristics.
Lab — build a misconception diagnostic
Setup: Use a calculator, spreadsheet or lab.py. No health or personal data are needed.
- Recreate the first frequency table and verify that row and column totals agree.
- Run
python lab.py probability. Expected output:2/13 = 15.3846%. - Construct the 10%-prevalence table and explain the changed denominator.
- Write a multiple-choice question with 90%, 5% and 15.38% as options. For each incorrect option, identify the quantity the learner has substituted.
- Create a transfer problem using a spam classifier or equipment alarm. State all assumptions and derive an answer before using it with learners.
Fault injection: Divide 90 by 100 and label the result as probability after a positive result. Use the table to identify why the label is wrong. Recovery: Restore the positive-column denominator of 585.
Evidence: Both tables, arithmetic, distractor explanations and transfer task. Cleanup: Retain only fictional examples and remove unsupported real-world claims.
Common pitfalls and teaching decisions
A rounded percentage should not be presented as exact. A small denominator can make a percentage unstable in measured data. This fixture uses exact constructed counts to teach the relationship; inference from observed samples would require additional uncertainty analysis.
Retrieval and reflection
Why is sensitivity not the answer to “given a positive result, what is the probability?”
Sensitivity conditions on the condition being present. The second question conditions on a positive result. They select different groups.
What changed when prevalence increased to 10%?
The balance of true and false positives changed, even though the stated sensitivity and false-positive rate stayed fixed. Rebuild the frequency table to see the effect.
Official references
OpenStax Introductory Statistics: probability · Python exact rational arithmetic.