Question

Suppose A and B are two independent events with probabilities \({\rm{P}}\left( {\rm{A}} \right) \ne 0\) and \({\rm{P}}\left( {\rm{B}} \right) \ne 0\). Let \({\rm{\bar A}}\) and \({\rm{\bar B}}\) be their complements. Which one of the following statements is False?
1. \({\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right){\rm{\;}} = {\rm{\;P}}\left( {\rm{A}} \right){\rm{P}}\left( {\rm{B}} \right)\)
2. \({\rm{P}}\left( {{\rm{A}}/{\rm{B}}} \right){\rm{\;}} = {\rm{\;P}}\left( {\rm{A}} \right)\)
3. \({\rm{P}}\left( {{\rm{A}} \cup {\rm{B}}} \right){\rm{\;}} = {\rm{\;P}}\left( {\rm{A}} \right){\rm{\;}} + {\rm{\;P}}\left( {\rm{B}} \right)\)
4. \({\rm{P}}\left( {{\rm{\bar A}} \cap {\rm{\bar B}}} \right){\rm{\;}} = {\rm{\;P}}\left( {{\rm{\bar A}}} \right){\rm{P}}\left( {{\rm{\bar B}}} \right)\)

Answer 1

Correct Answer - Option 3 : \({\rm{P}}\left( {{\rm{A}} \cup {\rm{B}}} \right){\rm{\;}} = {\rm{\;P}}\left( {\rm{A}} \right){\rm{\;}} + {\rm{\;P}}\left( {\rm{B}} \right)\)

Since A and B are independent

\(\begin{array}{l} {\rm{P\;}}\left( {{\rm{A}} \cap {\rm{B}}} \right){\rm{\;}} = {\rm{\;P\;}}\left( {\rm{A}} \right).{\rm{P\;}}\left( {\rm{B}} \right)\\ {\rm{P}}\left( {\frac{{\rm{A}}}{{\rm{B}}}} \right) = \frac{{{\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right)}}{{{\rm{P}}\left( {\rm{B}} \right)}} = \frac{{{\rm{P}}\left( {\rm{A}} \right).{\rm{P}}\left( {\rm{B}} \right)}}{{{\rm{P}}\left( {\rm{B}} \right)}} = {\rm{P}}\left( {\rm{A}} \right)\\ {\rm{P}}\left( {{\rm{\bar A}} \cap {\rm{\bar B}}} \right) = {\rm{P}}\left( {{\rm{\bar A}}} \right).{\rm{P}}\left( {{\rm{\bar B}}} \right) = \left[ {1 - {\rm{P}}\left( {\rm{A}} \right)} \right]\left[ {1 - {\rm{P}}\left( {\rm{B}} \right)} \right]\\ {\rm{P}}\left( {{\rm{A}} \cup {\rm{B}}} \right) = {\rm{P}}\left( {\rm{A}} \right) + {\rm{P}}\left( {\rm{B}} \right) - {\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right) \end{array}\)