Bayes' Theorem (Introduction)

Bayes' Theorem answers a distinctive kind of question: given that an OUTCOME has already been observed, what is the probability it came from one particular CAUSE among several possible causes? This reverses the usual direction of a conditional-probability question (which normally runs cause → effect); Bayes' Theorem runs effect → cause.

Setup — the Law of Total Probability. Suppose $E_1, E_2, \ldots, E_n$ are mutually exclusive and exhaustive events (they partition the sample space — exactly one of them must occur), and $A$ is any event that can occur along with one of the $E_i$'s. The overall probability of $A$ is then built up from every possible 'path' to it:

Bayes' Theorem. Given that $A$ has occurred, the (reversed) probability that it happened via cause $E_i$ specifically is:

$P(E_i|A)=\dfrac{P(E_i)\,P(A|E_i)}{\displaystyle\sum_{j=1}^{n}P(E_j)\,P(A|E_j)}$ …