Mark the correct alternative in the following question:

$If A and B are two independent events such that P (A) = 0.3 and P (A \cup B) = 0.5, then P (A | B) - P (B | A) = (a) \frac{2}{7} (b) \frac{3}{35} (c) \frac{1}{70} (d) \frac{1}{7}$

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Solution

$We have, P (A) = 0.3 and P (A \cup B) = 0.5 As, A and B are independent events So, P (A \cap B) = P (A) \times P (B) = 0.3 \times P (B) = 0.3 P (B) . . . . . (i) Also, P (A \cup B) = P (A) + P (B) - P (A \cap B) \Rightarrow 0.5 = 0.3 + P (B) - 0.3 P (B) [Using (i)] \Rightarrow 0.5 - 0.3 = 0.7 P (B) \Rightarrow 0.7 P (B) = 0.2 \Rightarrow P (B) = \frac{0.2}{0.7} \Rightarrow P (B) = \frac{2}{7} Using (i), we get P (A \cap B) = 0.3 \times \frac{2}{7} = \frac{6}{70} Now, P (A | B) - P (B | A) = \frac{P (A \cap B)}{P (B)} - \frac{P (A \cap B)}{P (A)} = \frac{(\frac{6}{70})}{(\frac{2}{7})} - \frac{(\frac{6}{70})}{0.3} = \frac{6 \times 7}{70 \times 2} - \frac{6}{70 \times 0.3} = \frac{3}{10} - \frac{2}{7} = \frac{21 - 20}{70} = \frac{1}{70}$

Hence, the correct alternative is option (c).

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