Conditional probability and Bayes' theorem

Conditional probability $P(A|B)$ is the chance of $A$ occurring given that $B$ has occurred:

$P(A|B)=\frac{P(A\cap B)}{P(B)},\quad P(B)>0$

When the “reverse” probabilities are known, we apply Bayes' theorem:

$P(A|B)=\frac{P(B|A)\,P(A)}{P(B|A)\,P(A)+P(B|\neg A)\,P(\neg A)}$

The denominator is $P(B)$ from the law of total probability. Bayes' theorem updates our knowledge: it combines the prior probability $P(A)$ with the “sensitivity” of the observation $P(B|A)$. Beware the rare-disease paradox — even an accurate test gives a low $P(A|B)$ when $P(A)$ is low.