Given a probability space \(\left(\Omega,\mathcal{F},\mathbb{P}\right)\) where \(\Omega\) is a sample space, \(\mathcal{F}\) is a \(\sigma\)-algebra on \(\Omega\) and \(\mathbb{P}\) is a probability measure, events \(A,B\in\mathcal{F}\) and \(\mathbb{P}\left(B\right)>0,\) the conditional probability of event \(A\) given that event \(B\) has occurred is:

\[\mathbb{P}\left(A\mid B\right)=\frac{\mathbb{P}\left(A\cap B\right)}{\mathbb{P}\left(B\right)}.\]

In general, \(\mathbb{P}\left(A\mid B\right)\neq\mathbb{P}\left(B\mid A\right).\)

If \(\mathbb{P}\left(B\right)>0,\) events \(A\) and \(B\) are independent events if and only if:

\[\mathbb{P}\left(A\mid B\right)=\mathbb{P}\left(A\right)\]

since, when \(A\) and \(B\) are independent, \(\mathbb{P}\left(A\mid B\right)=\frac{\mathbb{P}\left(A\right)\,\mathbb{P}\left(B\right)}{\mathbb{P}\left(B\right)}=\mathbb{P}\left(A\right),\) and conversely \(\mathbb{P}\left(A\mid B\right)=\mathbb{P}\left(A\right)\) rearranges to \(\mathbb{P}\left(A\cap B\right)=\mathbb{P}\left(A\right)\,\mathbb{P}\left(B\right).\)

Conditional probability is equivalent to reducing the sample space by replacing events with intersections between themselves and the event that has occurred and rescaling probabilities.

Given events \(A,B,C\in\mathcal{F},\) events \(A\) and \(B\) are conditionally independent given \(C\) if and only if:

\[\mathbb{P}\left(A\cap B\mid C\right)=\mathbb{P}\left(A\mid C\right)\,\mathbb{P}\left(B\mid C\right)\]

or, equivalently when \(\mathbb{P}\left(B\cap C\right)>0:\)

\[\mathbb{P}\left(A\mid B,C\right)=\mathbb{P}\left(A\mid C\right)\]

where \(\mathbb{P}\left(A\mid B,C\right)\) denotes \(\mathbb{P}\left(A\mid B\cap C\right).\)

Conditional independence relationships can be written as \(A\perp\!\!\!\perp B\mid C.\)

The theorem of total probability states that, given a partition on \(\Omega\) containing \(k\) events \(E_1,\ldots,E_k\in\mathcal{F}\) with \(\mathbb{P}\left(E_i\right)>0,\) the probability of an event \(A\in\mathcal{F}\) occurring is given by:

\[\mathbb{P}\left(A\right)=\sum_{i=1}^k \mathbb{P}\left(A\mid E_i\right)\,\mathbb{P}\left(E_i\right).\]

Bayes’ theorem states that, given a partition on \(\Omega\) containing \(k\) events \(E_1,\ldots,E_k,\) \(\mathbb{P}\left(E_i\right)>0\) and \(\mathbb{P}\left(A\right)>0,\) then:

\[\begin{align} \mathbb{P}\left(E_i\mid A\right)&=\frac{\mathbb{P}\left(A\mid E_i\right)\,\mathbb{P}\left(E_i\right)}{\mathbb{P}\left(A\right)}\\ &=\frac{\mathbb{P}\left(A\mid E_i\right)\,\mathbb{P}\left(E_i\right)}{\sum_{j=1}^k \mathbb{P}\left(A\mid E_j\right)\,\mathbb{P}\left(E_j\right)}. \end{align}\]