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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
\[ \frac{4}{15}\]
\[ \frac{8}{45}\]
\[ \frac{1}{3} \]
\[ \frac{2}{9}\]
Solution
\[\text{ We have } , \]
\[P\left( A \right) = \frac{3}{5} \text{ and } P\left( B \right) = \frac{4}{9}\]
\[\text{ As, A and B are independent events } \]
\[\text{ So} , P\left( A \cap B \right) = P\left( A \right) \times P\left( B \right)\]
\[ = \frac{3}{5} \times \frac{4}{9}\]
\[ = \frac{12}{45}\]
\[ = \frac{4}{15}\]
\[\text{ Also } , P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]
\[ = \frac{3}{5} + \frac{4}{9} - \frac{4}{15}\]
\[ = \frac{27 + 20 - 12}{45}\]
\[ = \frac{35}{45}\]
\[ = \frac{7}{9}\]
\[\text{ Now } , \]
\[P\left( \overline{A} \cap \overline{B} \right) = P\left( \overline{A \cup B} \right)\]
\[ = 1 - P\left( A \cup B \right)\]
\[ = 1 - \frac{7}{9}\]
\[ = \frac{2}{9}\]
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