Bayes' Theorem

Topics: Probability

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(theorem)

Let $(B_i{)}_{i\in \mathbb{N}}$ be a partition of a sample space and let $A$ be an event. Then:

$$\begin{array}{rl}\mathbb{P}(B_i\mid A)& =\frac{\mathbb{P}(A\mid B_i)\mathbb{P}(B_i)}{\mathbb{P}(A)}\\ & =\frac{\mathbb{P}(A\mid B_i)\mathbb{P}(B_i)}{{\sum }_{i\in \mathbb{N}}\mathbb{P}(A\mid B_i)\mathbb{P}(B_i)}\end{array}$$

See conditional probability. The second equality follows after the total probability formula.

Basically, Bayes’ theorem describes the probability of an event based on prior knowledge of conditions that might be related to the event.