Question

Mark the correct alternative in the following question: 

\[\text{ If A and B are two independent events such that}  P\left( A \right) = 0 . 3 \text{ and } P\left( A \cup B \right) = 0 . 5, \text{ then } P\left( A|B \right) - P\left( B|A \right) = \]

 

 

Options

• \[ \frac{2}{7}\]

• \[ \frac{3}{35}\]

• \[ \frac{1}{70} \]

• \[ \frac{1}{7}\]

Answer 1

\[\text{ We have } , \]

\[P\left( A \right) = 0 . 3 \text{ and } P\left( A \cup B \right) = 0 . 5\]

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

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

\[ = 0 . 3 \times P\left( B \right)\]

\[ = 0 . 3P\left( B \right) . . . . . \left( i \right)\]

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

\[ \Rightarrow 0 . 5 = 0 . 3 + P\left( B \right) - 0 . 3P\left( B \right) \left[ \text{Using } \left( i \right) \right]\]

\[ \Rightarrow 0 . 5 - 0 . 3 = 0 . 7P\left( B \right)\]

\[ \Rightarrow 0 . 7P\left( B \right) = 0 . 2\]

\[ \Rightarrow P\left( B \right) = \frac{0 . 2}{0 . 7}\]

\[ \Rightarrow P\left( B \right) = \frac{2}{7}\]

\[\text{ Using } \left( i \right), \text{ we get } \]

\[P\left( A \cap B \right) = 0 . 3 \times \frac{2}{7} = \frac{6}{70}\]

\[\text{ Now } , \]

\[P\left( A|B \right) - P\left( B|A \right) = \frac{P\left( A \cap B \right)}{P\left( B \right)} - \frac{P\left( A \cap B \right)}{P\left( A \right)}\]

\[ = \frac{\left( \frac{6}{70} \right)}{\left( \frac{2}{7} \right)} - \frac{\left( \frac{6}{70} \right)}{0 . 3}\]

\[ = \frac{6 \times 7}{70 \times 2} - \frac{6}{70 \times 0 . 3}\]

\[ = \frac{3}{10} - \frac{2}{7}\]

\[ = \frac{21 - 20}{70}\]

\[ = \frac{1}{70}\]