Bayes' Theorem (Two Hypotheses)

Bayes' theorem turns "how often does this evidence show up when the hypothesis is true?" into "how likely is the hypothesis now that I have seen the evidence?" The denominator is the total probability of the evidence from both branches — true positives plus false positives. Thomas Bayes never published it; his friend Richard Price found the essay among his papers and read it to the Royal Society in 1763, two years after Bayes died. Laplace rediscovered the idea independently and used it on everything from Saturn's mass to the sex ratio at birth, and Alan Turing's Banburismus at Bletchley Park was Bayesian updating in industrial quantities.

The famous trap is ignoring the base rate. Take a disease affecting 1% of people and a test with 99% sensitivity and a 5% false-positive rate: a positive result gives P = (0.99 × 0.01) / (0.99 × 0.01 + 0.05 × 0.99) = 0.0099/0.0594 = 1/6 ≈ 0.167. Only about one positive in six is a real case, because the 99% of healthy people generate far more false alarms than the 1% of sick people generate true ones. When Casscells and colleagues put a version of this to Harvard medical staff in 1978, most answered 95%. Enter every rate as a decimal from 0 to 1, or switch the unit to %.