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 \[B\cap C=B-(A\cap B\cap \overline{C})-(\overline{A}\cap B\cap \overline{C})\] \[\therefore \]\[P(B\cap C)=P(B)-P(A\cap B\cap \overline{C})\] \[-P(\overline{A}\cap B\cap \overline{C})\] \[=\frac{3}{4}-\frac{1}{3}-\frac{1}{3}=\frac{3}{4}-\frac{2}{3}=\frac{1}{12}\]You need to login to perform this action.
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