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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
\[ \frac{2}{7}\]
\[ \frac{3}{35}\]
\[ \frac{1}{70} \]
\[ \frac{1}{7}\]
Solution
\[\text{ We have } , \]
\[P\left( A \right) = 0 . 3 \text{ and } P\left( A \cup B \right) = 0 . 5\]
\[\text{ As, A and B are independent events} \]
\[\text{ So } , P\left( A \cap B \right) = P\left( A \right) \times P\left( B \right)\]
\[ = 0 . 3 \times P\left( B \right)\]
\[ = 0 . 3P\left( B \right) . . . . . \left( i \right)\]
\[\text{ Also} , P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]
\[ \Rightarrow 0 . 5 = 0 . 3 + P\left( B \right) - 0 . 3P\left( B \right) \left[ \text{Using } \left( i \right) \right]\]
\[ \Rightarrow 0 . 5 - 0 . 3 = 0 . 7P\left( B \right)\]
\[ \Rightarrow 0 . 7P\left( B \right) = 0 . 2\]
\[ \Rightarrow P\left( B \right) = \frac{0 . 2}{0 . 7}\]
\[ \Rightarrow P\left( B \right) = \frac{2}{7}\]
\[\text{ Using } \left( i \right), \text{ we get } \]
\[P\left( A \cap B \right) = 0 . 3 \times \frac{2}{7} = \frac{6}{70}\]
\[\text{ Now } , \]
\[P\left( A|B \right) - P\left( B|A \right) = \frac{P\left( A \cap B \right)}{P\left( B \right)} - \frac{P\left( A \cap B \right)}{P\left( A \right)}\]
\[ = \frac{\left( \frac{6}{70} \right)}{\left( \frac{2}{7} \right)} - \frac{\left( \frac{6}{70} \right)}{0 . 3}\]
\[ = \frac{6 \times 7}{70 \times 2} - \frac{6}{70 \times 0 . 3}\]
\[ = \frac{3}{10} - \frac{2}{7}\]
\[ = \frac{21 - 20}{70}\]
\[ = \frac{1}{70}\]
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