Conditional probability

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$\newcommand{\E}[1]{ {\mathbb{E}{\left[{#1}\right]} } }$$\newcommand{\Mdm}[1]{\text{Mdm}{\left({#1}\right) } }$$\newcommand{\Var}[1]{\text{Var}{\left({#1}\right) } }$$\newcommand{\ncr}[2]{ \vphantom{C}^{#1}\!C_{#2} }$

Definition

Let $(S,\Omega,\mathbb{P})$ be a probability space and let $C\in\Omega$ be an event such that $\P{C}>0$, then we may define a new probability space, $(C,\Omega_C,\mathbb{P}_C:\Omega_C\rightarrow\mathbb{R})$ where:

• $\Omega_C$ is the trace $\sigma$-algebra, $\Omega_C:\eq{\left\{ C\cap A\ \middle\vert\ A\in\Omega\right\} }$, and
• $\mathbb{P}_C:\Omega_C\rightarrow\mathbb{R}$ defined by $$\P[_C]{A}:\eq \frac{\P{A} }{\P{C} }$$
  • Notice that as $A\in\Omega_C$ that $\exists A'\in\Omega[A\eq A'\cap C]$ and thus $\P{A'\cap C}\eq\P{A}$ as of course a $\sigma$-algebra