Question

Mark the correct alternative in the following question: 

\[\text{ If A and B are two independent events with } P\left( A \right) = \frac{3}{5} \text{ and } P\left( B \right) = \frac{4}{9}, \text{ then } P\left( \overline{A} \cap B \right) \text{ equals } \]

Options

• \[ \frac{4}{15}\]

• \[ \frac{8}{45}\]

• \[ \frac{1}{3} \]

• \[ \frac{2}{9}\]

Answer 1

\[\text{ We have } , \]

\[P\left( A \right) = \frac{3}{5} \text{ and }  P\left( B \right) = \frac{4}{9}\]

\[\text{ As, A and B are independent events } \]

\[\text{ So} , P\left( A \cap B \right) = P\left( A \right) \times P\left( B \right)\]

\[ = \frac{3}{5} \times \frac{4}{9}\]

\[ = \frac{12}{45}\]

\[ = \frac{4}{15}\]

\[\text{ Also } , P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]

\[ = \frac{3}{5} + \frac{4}{9} - \frac{4}{15}\]

\[ = \frac{27 + 20 - 12}{45}\]

\[ = \frac{35}{45}\]

\[ = \frac{7}{9}\]

\[\text{ Now } , \]

\[P\left( \overline{A} \cap \overline{B} \right) = P\left( \overline{A \cup B} \right)\]

\[ = 1 - P\left( A \cup B \right)\]

\[ = 1 - \frac{7}{9}\]

\[ = \frac{2}{9}\]