Question

If A⊂B with Pr(B) > 0, what is the value of \({\bf{Pr}}\left( {{\bf{A}}|{\bf{B}}} \right)\) ?

Short answer

If A⊂B with Pr (B) > 0, then the value of\(Pr\left( {A\left| B \right.} \right)\)is:

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

Step-by-step solution

Given information

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

Finding the conditional probability Pr (A|B)

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:

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