91Ó°ÊÓ

Independence is a state in which a person, a nation, a country, or a state's people and population, or a portion of them, have self-government and, in most cases, sovereignty over their area. The status of a dependent territory is the polar opposite of independence.

Multiplication Property: Two events A and B are independent if and only if

\(P(A \cap B) = P(A) \cdot P(B)\)

As we are asked, we need to use the proposition given above. From exercise\({\rm{13}}\), we have

\(\begin{array}{l}P\left( {{A_1} \cap {A_2}} \right) = 0.11\\P\left( {{A_1} \cap {A_3}} \right) = 0.05\\P\left( {{A_2} \cap {A_3}} \right) = 0.07\end{array}\)

as well as the probabilities of event\({{\rm{A}}_{\rm{1}}}{\rm{,}}{{\rm{A}}_{\rm{2}}}\), and \({{\rm{A}}_{\rm{3}}}\)

\(\begin{array}{c}P\left( {{A_1}} \right) \cdot P\left( {{A_2}} \right) &=& 0.22 \cdot 0.25\\ &=& 0.055\\P\left( {{A_1}} \right) \cdot P\left( {{A_3}} \right) &=& 0.22 \cdot 0.28\\ &=& 0.0616\\P\left( {{A_2}} \right) \cdot P\left( {{A_3}} \right) &=& 0.25 \cdot 0.28\\ &=& 0.07\end{array}\)

From the multiplication rule, the following is true

\(P\left( {{A_1} \cap {A_2}} \right) = 0.11 \ne 0.055 = P\left( {{A_1}} \right) \cdot P\left( {{A_2}} \right)\)

The following is holds

\(P\left( {{A_1} \cap {A_3}} \right) = 0.05 \ne 0.0616 = P\left( {{A_1}} \right) \cdot P\left( {{A_3}} \right)\)

Therefore, \({{\rm{A}}_{\rm{1}}}\)and \({{\rm{A}}_{\rm{3}}}\)are dependent.

The following is true

\(\begin{array}{c}P\left( {{A_2} \cap {A_3}} \right) &=& 0.07\\ &=& 0.07\\ &=& P\left( {{A_2}} \right) \cdot P\left( {{A_3}} \right)\end{array}\)

The following is true

Therefore, \({{\rm{A}}_1}\)and \({{\rm{A}}_{\rm{3}}}\)are independent.