91Ó°ÊÓ

\(A \subset B\) with \(P\left( B \right) > 0\)

The conditional probability of the event A given that the event B has occurred, denoted Pr (A|B) .

The formula is stated as follows:

\({\bf{Pr}}\left( {{\bf{A|B}}} \right){\bf{ = }}\frac{{{\bf{Pr}}\left( {{\bf{A}} \cap {\bf{B}}} \right)}}{{{\bf{Pr}}\left( {\bf{B}} \right)}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;...\left( {\bf{1}} \right)\)

Where,\(\Pr \left( B \right) > 0\;\;\;.\)

Since,\(A \subset B\); therefore,

\(\Pr \left( {A \cap B} \right) = \Pr \left( A \right)\)

Thus, equation (1) becomes:

\(\begin{aligned}{}\Pr \left( {A|B} \right) &= \frac{{\Pr \left( {A \cap B} \right)}}{{\Pr \left( B \right)}}\\ &= \frac{{\Pr \left( A \right)}}{{\Pr \left( B \right)}}\end{aligned}\)

Therefore, the required value of the given expression is:

\(\Pr \left( {A|B} \right) = \frac{{\Pr \left( A \right)}}{{\Pr \left( B \right)}}\)