Bayes on a population grid

$P(A\mid B)$ is the probability of A given that B happened: restrict the world to B, then ask how much of it is A. By definition $P(A\mid B) = P(A \cap B)/P(B)$.

Write the joint probability both ways, $P(A\cap B) = P(A\mid B)P(B) = P(B\mid A)P(A)$, and divide: that is Bayes' theorem. It flips a conditional around: from 'how likely is this evidence if the hypothesis is true' to 'how likely is the hypothesis given the evidence'.

The classic trap is ignoring the base rate. A 99%-accurate detector for something that affects 1 in 1,000 people still produces mostly false positives. The simulation lets you see it as a population grid.