Q. $If\ P(B) = \frac{3}{4},\ P\left( A \cap B \cap \overline{C} \right) = \frac{1}{3}\ and\ P\left( \overline{A} \cap B \cap \overline{C} \right) = \frac{1}{3},\ then\ P(B \cap C)\ is$
A) $\frac{1}{12}$
B) $\frac{1}{6}$
C) $\frac{1}{15}$
D) $\frac{1}{9}$
Explanation
(a)
$P\left( \overline{A} \cap \left( B \cap \overline{C} \right) \right\} = P\left( B \cap \overline{C} \right) - P(A \cap B \cap \overline{C})$
$= P(B) - P(B \cap C) - P(A \cap B \cap \overline{C})$
$\Rightarrow \ - P\left( \overline{A} \cap B \cap \overline{C} \right) - P(A \cap B \cap \overline{C}) + P(B) = P(B \cap C)$
$\Rightarrow \ P(B \cap C) = \frac{3}{4} - \frac{1}{3} - \frac{1}{3} = \frac{1}{12}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.