intro practice · coding problem
Apply Bayes' rule
A test has 99% sensitivity and 95% specificity. Disease prevalence is 1%. A patient tests positive. Using Bayes' rule, compute P(disease | positive) to 2 decimal places.
The mechanism
Probability is a bookkeeping system for uncertainty. Conditional probability restricts the sample space; Bayes flips cause and effect — from 'how likely is the evidence given the hypothesis' to 'how likely is the hypothesis given the evidence'.
Expected evidence
≈ 0.17 (17%). Most positives are false alarms when the base rate is low.
Try this drill in the practice workspace →
Hints when you need them
- P(D|+) = P(+|D)·P(D) / P(+).
- P(+) = P(+|D)·P(D) + P(+|¬D)·P(¬D).
Watch for these mistakes
- Treating independent events as mutually exclusive
- Confusing P(A|B) with P(B|A) (the prosecutor's fallacy)
- Adding probabilities of overlapping events without inclusion-exclusion