> [!NOTE] Definition (Conditional Probability) > Let $(\Omega, \mathcal{F},\mathbb{P})$ be a [[Probability Space|probability space]]. Let $B\in F$ such that $\mathbb{P}(B)>0.$ For $A\in \mathcal{F},$ the conditional probability of $A$ given $B$ is given by $\mathbb{P}(A\mid B)= \frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}.$ # Properties By [[Conditional Probability Defines Probability Space]], if $B\in \mathcal{F}$ such that $\mathbb{P}(B)>0$ then $(\Omega,\mathcal{F},P_{B})$ is a probability space where $P_{B}(A)= \mathbb{P}(A\mid B)$: that is $P_{B}$ is a [[Probability Measure|probability measure]] on $(\Omega, \mathcal{F}).$ # Applications The [[Chain Rule for Probability]] asserts that $\mathbb{P}(A_{1} \cap \dots \cap A_{n})= \mathbb{P}(A_{1})\mathbb{P}(A_{2}\mid A_{1})\mathbb{P}(A_{3}\mid A_{1} \cap A_{2})\dots \mathbb{P}(A_{n}\mid A_{1} \cap A_{2} \cap \dots \cap A_{n-1})$ where $A_{1},\dots ,A_{n}$ is a finite sequence of events such that $\mathbb{P}(A_{1} \cap \dots \cap A_{n}).$ Note that the last condition gives that the conditional probabilities are well-defined i.e. $\mathbb{P}(A_{1} \cap \dots \cap A_{k})>0$ for all $k= 1,2,\dots,n-1$ since $A_{1} \cap \dots A_{n} \subset A_{1} \cap \dots \cap A_{k}.$ The [[Law of Total Probability]] asserts that if $\mathbb{B}$ forms a partition of the sample space, then $\mathbb{P}(A) = \sum_{B\in \mathbb{B} }\mathbb{P}(A \cap B) = \sum_{B\in \mathbb{B} }\mathbb{P}(A)\mathbb{P}(B \mid A).$ [[Bayes' Theorem]] asserts that for all $A\in \mathcal{F}$ such that $\mathbb{P}(A)>0,$ $\mathbb{P}(B\mid A)= \frac{\mathbb{P}(A \mid B) \mathbb{P}(B)}{\sum_{D\in \mathbb{B}}\mathbb{P}(A\mid D)\mathbb{P}(D)}.$