Conditional probability

[ilmath]\newcommand{\P}[2][]{\mathbb{P}#1{\left[{#2}\right]} } \newcommand{\Pcond}[3][]{\mathbb{P}#1{\left[{#2}\!\ \middle\vert\!\ {#3}\right]} } \newcommand{\Plcond}[3][]{\Pcond[#1]{#2}{#3} } \newcommand{\Prcond}[3][]{\Pcond[#1]{#2}{#3} }[/ilmath]

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

Definition

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

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