A test is 90% accurate and you test positive for a one-in-a-hundred disease. Your odds of actually having it aren’t 90% — they’re about 8%, because the rare disease is drowned by false alarms. Slide the prevalence (or tap) and watch the true positives appear or vanish in the crowd.
Bayes' rule on a screening test with sensitivity 0.90 and specificity 0.90. For prevalence p, the posterior P(sick | positive) = (0.90·p) / (0.90·p + 0.10·(1−p)). At p=0.01 the posterior is only ~8% — the base rate is so low that the 10% false-positive rate over the healthy majority swamps the true positives; raise the prevalence and the posterior climbs. A 100-person grid shows the true/false positives live. A fail-loud self-check throws unless the posterior is below 0.15 at p=0.01 and above 0.8 at p=0.5.
Fixed 90/90 test accuracy and a 100-person illustrative grid; real tests trade sensitivity against specificity and prevalence is estimated. The Bayes posterior is computed exactly.