Conditional Probability

Definition (Conditional probability)

If $A$ and $B$ are events with $\mathbf{P}(B)>0$, then the conditional probability of $A$ given $B$, denoted $\mathbf{P}(A|B)$, is defined as

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

$\mathbf{P}(\cdot |B)$ satisfies the axioms of probability, for a fixed $B$. In general, $\mathbf{P}(A|\cdot )$ does not satisfy the axioms of probability, for fixed $A$. Also, in general, $\mathbf{P}(A|B)\ne \mathbf{P}(B|A)$.

Theorem (Chain rule)

For two events $A$ and $B$,

$$\mathbf{P}(A\cap B)=\mathbf{P}(B|A)\mathbf{P}(A).$$

For any events $A_1,\dots ,A_n$ with positive probabilities,

$$\begin{array}{rl}\mathbf{P}(A_1\cap \cdots \cap A_n)& =\mathbf{P}(A_1)\mathbf{P}(A_2|A_1)\mathbf{P}(A_2|A_1,A_2)\cdots \mathbf{P}(A_n|A_1,\dots ,A_{n-1})\\ & =\prod _{k=1}^n\mathbf{P}\left(A_k|\bigcap _{j=1}^{k-1}{A}_j\right).\end{array}$$