Bayes' theorem

In probability theory, Bayes’ theorem relates the conditional probability of an event $B$ given an event $A$ occurs. In the single event case:

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

In the multiple event case, where we have multiple disjoint sets $B_1,\cdots ,B_k$, then the probability of an event $B_i$ given event $A$ occurs is given by:

$$P(B_i|A)=\frac{P(B_i)P(A|B_i)}{{\sum }_{k=0}^{n}P(B_k)P(A|B_k)}$$