Whole Earth Curriculum WEC

Lesson 13 — Independence and Conditional Probability: Reading Screening Tests

Learners meet conditional probability P(A|B), define independence, and use both to read a screening test honestly — discovering that with a rare disease, most positives can be false. They reflect on what that means for who is tested, how results are communicated, and whether testing is fair.

D05 P3: Intellectual & Cognitive Awareness D05.S4 55 minutes Draft

Objectives

Essential question

How does new information change a probability, and why can a "positive" screening test still mean a person is probably fine?

conditional probabilityindependencesensitivityspecificityfalse positivefalse negativebase rateprevalence

Materials

Standard materials

Low-tech / no-cost

Enriched / lab & device

Works in different contexts

A two-by-two contingency table for a screening test on a rare disease, showing true positives, false positives, true negatives, and false negatives, with the conditional probability P(disease given positive test) computed beside it, in grayscale-printable labels
A two-by-two contingency table for a screening test on a rare disease, showing true positives, false positives, true negatives, and false negatives, with the conditional probability P(disease given positive test) computed beside it, in grayscale-printable labels

Lesson 13 — Independence and Conditional Probability: Reading Screening Tests

Summary

Learners meet conditional probability P(A|B), define independence, and use both to read a screening test honestly — discovering that when a disease is rare, most positive results can be false. They reflect on what that means for who gets tested, how results are communicated, and whether testing is fair.

Objectives

  • Understand independence and conditional probability and use them to interpret real situations such as screening tests and risk. (D05.S4.10.01)

Connection

A screening test is 99% accurate. You take it, and it comes back positive. Surely you almost certainly have the disease — right? Not necessarily. If the disease is rare — say 1 in 1,000 people have it — then even a very good test will “find” far more healthy people by mistake than truly sick people, simply because there are so many more healthy people to mistake. The math is a humbling, life-relevant truth: new information changes a probability, but the base rate matters just as much as the test. Reading this honestly protects people from needless fear, needless treatment, and unfair labeling — and it is exactly what conditional probability is for.

Materials

  • Screening-test worksheet
  • Math journal

Preparation

  • Copy or draw the screening-test worksheet.
  • Retrieval: from Lesson 12, two-way tables and the multiplication rule; from Grade 9, conditional relative frequencies (D05.S4.09.02). Today we formalize P(A|B) and independence.
  • Prepare the rare-disease worked table.

Facilitator note

This lesson is written to the learner (“you”). The idea to land: conditional probability P(A|B) = P(A and B) / P(B) measures how knowing B changes the chance of A; independence means P(A|B) = P(A); and for screening tests the key insight is that with a rare condition, most positives are false — so the prevalence (base rate) matters as much as the test’s accuracy. Teach the computation with worked examples (S-011), and let the 2×2 table carry the reasoning.

The intellectual lens: the formula is just “restrict to the cases where B is true, and ask what fraction are A.” The critical-thinking lens: always ask “given what?” — a probability without its condition is incomplete. The ethics lens: a false alarm is not a harmless number — it is fear, cost, and sometimes treatment a person did not need; honest communication of risk is a duty, not a nicety. The egalitarian lens: who gets screened, who can afford follow-up, and how a “positive” label follows a person are questions of fairness — the same arithmetic lands differently on different people. The technology lens: screening, prediction, and risk scores are everywhere in health and beyond; the base-rate lesson applies to any “it flagged you” system. Preview: Lesson 14 ends the unit with expected value — the decision side of risk.

Procedure

  1. Recall (5 min). From Lesson 12, what is P(A and B) for two independent events? From Grade 9, what is a two-way table? Today we ask: how does knowing one thing change the chance of another?
  2. Meet conditional probability (10 min). P(A|B) is read “the probability of A given B” and equals P(A and B) / P(B). Worked example: from a deck, P(king | heart) = P(king and heart) / P(heart) = (1/52)/(13/52) = 1/13. Two events are independent if knowing one does not change the other: P(A|B) = P(A).
  3. Read a screening test (20 min). Work the table for a disease with prevalence 1 in 1,000 in a town of 10,000, and a test with sensitivity 99% (catches 99% of the sick) and specificity 99% (clears 99% of the well):
    • Sick: 10; well: 9,990.
    • True positives: 99% of 10 ≈ 10. False negatives: ≈ 0.
    • False positives: 1% of 9,990 ≈ 100. True negatives: ≈ 9,890.
    • So P(disease | positive) ≈ 10 / (10 + 100) ≈ 9%. Even with a positive result, the person is probably fine — because the disease is rare. Recompute with a common disease (prevalence 10%) and see the answer change.
  4. Reflect (10 min). In your journal, answer: why can a positive test still mean “probably fine”? Who bears the cost of a false alarm — the person, the family, the system? What would make testing fairer?
  5. Close (5 min). Say in one sentence what P(A|B) measures and why the base rate matters as much as the test.

Differentiation

  • Support: Fill the 2×2 table with counters first, then translate counts to fractions.
  • Extension: Compute P(well | negative) for the same table and explain what a negative result tells you when the test is very sensitive.

Assessment

  • Formative (peer + self): Can the learner compute a conditional probability from a 2×2 table, test for independence, and explain the base-rate effect in a screening scenario?
  • Portfolio artifact (unit): The completed screening-test worksheet with the written reflection, added to the data-and-chance toolkit.

Home connection

Ask someone about a test or a prediction they have seen — medical, weather, or a “risk score.” Talk together about what the number was conditioned on, and whether the base rate was part of the story.

Resources