Bayes' theorem: medical test

Set how common a condition is and how accurate a test is, then see through a tree diagram and 1000 people how likely a positive result really means having the condition.

Prevalence P(C)

Sensitivity P(+ | C)

Specificity P(− | C′)

Condition on result

Try this

With prevalence 1 %, sensitivity 90 % and specificity 95 %, find P(C | +). Why is it so much lower than 90 %?
Keep the test fixed and raise the prevalence to 20 %. Watch how the number of false positives compares with true positives.
At prevalence 1 %, raise specificity from 95 % to 99.5 %, then reset and raise sensitivity to 99.5 % instead. Which change improves P(C | +) more?

Common error: confusing P(+ | C) with P(C | +); draw the tree, multiply along each branch, then divide the branch you want by the total of all branches giving that result.

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More on 4: Venn diagrams and conditional probability · Discrete random variables · Continuous random variables · Standard deviation and outliers

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