Enter a prevalence, a sensitivity, and a false-positive rate to see what a positive test result is actually worth - as a probability and as whole people out of ten thousand.
A positive result is not the same as having the condition.
The test finds 90 genuine cases, but also flags 495 healthy people. Since a positive could be either, the chance it is real is just 15.4%. Raise the prevalence and watch it climb - the base rate, not the accuracy of the test, is doing most of the work.
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Derive Bayes' theorem from the definition of conditional probability, then work the base-rate example that fools almost everyone, twice: once with the formula and once by pure counting.
Build probability from the ground up: possible worlds, the sample space, the two basic axioms, and the addition and multiplication rules, each derived rather than asserted, with worked numeric examples.