Definitions and useful formulas

The probability of conditions \(A\) and \(B\) occuring at the same time

\[P(A,B) \equiv P(A \text{ and } B)\]

Addition Rule

The probability of \(A\) or \(B\) occuring is given by

\[P(A \cup B) = P(A)+P(B)-P(A,B)\]

The conditional Probability, of an event \(A\) given a condition \(B\), denoted \(P(A|B)\) is defined as

\[P(A|B) = \frac{P(A,B)}{P(B)}\]

\[P(A,B) = P(A|B) P(B)\]

\[\begin{align} P(A) &= \sum\limits_{i=0}^N P(A,B_i) \\ &= \sum\limits_{i=0}^N P(A|B_i)P(B_i) \end{align}\]

In the continuous case we have

\[\begin{align} P(A) &= \int d\theta \ P(A,\theta) \\ &= \int d\theta \ P(A|\theta)P(\theta) \end{align}\]

\[P(A|B) = \frac{P(B|A)P(A)}{P(B)}\]