question_answer

    If\[P(B)=\frac{3}{4},P(A\cap B\cap \overline{C})=\frac{1}{3}\]and\[P(\overline{A}\cap B\cap \overline{C})=\frac{1}{3},\]then\[P(B\cap C)\]is

A)  1/12                                     

B)  1/6

C)  1/15                                     

D)  1/9

Correct Answer: A

Solution :

                Key Idea: \[P(A\cap \overline{B})=P(A)-P(A\cap B)\] Given, \[P(B)=\frac{3}{4},P(A\cap B\cap \overline{C})=\frac{1}{3}\] and       \[P\{\overline{A}\cap B\cap \overline{C}\}=\frac{1}{3}\] Now, \[P\{\overline{A}\cap (B\cap \overline{C})\}=P(B\cap \overline{C})\]                                                 \[-P(A\cap B\cap \overline{C})\] \[=P(B)-P(B\cap C)-P(A\cap B\cap \overline{C})\] \[\Rightarrow \] \[-P(\overline{A}\cap B\cap \overline{C})-P(A\cap B\cap \overline{C})+P(B)\]                 \[=P(B\cap C)\] \[\Rightarrow \] \[P(B\cap C)=\frac{3}{4}-\frac{1}{3}-\frac{1}{3}=\frac{3}{4}-\frac{2}{3}=\frac{1}{12}\] Alternative Method \[A\cap B\cap \overline{C}\] \[P(B\cap C)\]is                 \[\overline{A}\cap B\cap \overline{C}\]is                 From figure \[B\cap C=B-(A\cap B\cap \overline{C})-(\overline{A}\cap B\cap \overline{C})\] \[\therefore \]\[P(B\cap C)=P(B)-P(A\cap B\cap \overline{C})\]                                                 \[-P(\overline{A}\cap B\cap \overline{C})\] \[=\frac{3}{4}-\frac{1}{3}-\frac{1}{3}=\frac{3}{4}-\frac{2}{3}=\frac{1}{12}\]