Opening
A screening test can detect ninety percent of a rare condition and still be wrong most of the time when it says somebody has it. There is nothing mystical about this. There are just a great many healthy people.
Teaching narrative
Separate sensitivity, specificity, prevalence, and positive predictive value. The first two describe a test conditional on the actual condition status. The last describes condition status conditional on a positive result. A test does not become infallible just because its sensitivity is impressive.
Count expected positives among those with the condition and those without it. When prevalence is one percent, the much larger population without the condition can generate a substantial number of false positives, even at a five-percent false-positive rate.
Expected value is also a weighted average, not a promise about a single observation. In a game you might expect a one-dollar gain despite outcomes that are always either positive ten dollars or negative eight dollars.
Binomial and Poisson models describe different count settings. Both are useful only when their assumptions resemble the actual process: independent fixed-probability trials for the simple binomial case, and a suitable event-rate process for the basic Poisson model.
Before the experiment
Out of 10,000 screenings, will the expected number of false positives exceed true positives?
Live demonstration
Set prevalence to 1%, sensitivity 90%, specificity 95% in the Bayesian laboratory. In R, compute dbinom and dpois for an explicitly specified model.
Exit question
Explain why a positive predictive value cannot be given without a prevalence assumption.