Bayes Theorem / False Positive Explorer

Sensitivity and specificity describe the test; prevalence describes the population before testing. Bayes theorem combines both. When prevalence is low, even a small false-positive rate can create many false alarms because most tested people do not have the condition. That is why a positive screening result often needs confirmation, while the same test can be highly persuasive in a high-risk group.

Bayes theorem is a rule for changing your mind with evidence. You start with a prior — your best estimate before testing, which here is the prevalence — and the result nudges it up or down into a posterior. A single positive rarely jumps you straight to certainty; it multiplies your starting odds rather than overwriting them. The takeaway: before trusting any result, first ask how likely the thing was in the first place.

Confusing P(positive | disease) with P(disease | positive) has a courtroom name: the prosecutor's fallacy. In the 1999 trial of Sally Clark, a jury heard that two cot deaths in one family had a 1-in-73-million chance and treated it as the chance she was innocent. Those are different numbers — the second ignores how rare the alternative (a double murder) also is. Her conviction was overturned in 2003, and the Royal Statistical Society publicly condemned the misuse.