Editing Bayes' theorem (section)

====Extended form====
Often, for some [[partition of a set|partition]] {''A<sub>j</sub>''} of the [[sample space]], the [[Sample space|event space]] is given in terms of ''P''(''A<sub>j</sub>'') and ''P''(''B''&nbsp;|&nbsp;''A<sub>j</sub>''). It is then useful to compute ''P''(''B'') using the [[law of total probability]]:

<math>P(B)=\sum_{j}P(B \cap A_j),</math>

Or (using the multiplication rule for conditional probability),<ref>{{Cite web |title=Bayes Theorem - Formula, Statement, Proof {{!}} Bayes Rule |url=https://www.cuemath.com/data/bayes-theorem/ |access-date=2023-10-20 |website=Cuemath |language=en}}</ref>

:<math>P(B) = {\sum_j P(B| A_j) P(A_j)},</math>
:<math>\Rightarrow P(A_i| B) = \frac{P(B| A_i) P(A_i)}{\sum\limits_j P(B| A_j) P(A_j)}\cdot</math>

In the special case where ''A'' is a [[binary variable]]:

:<math>P(A| B) = \frac{P(B| A) P(A)}{ P(B| A) P(A) + P(B| \neg A) P(\neg A)}\cdot</math>