question_answer

Two events A and B are such that \[P(A)=\frac{1}{2},\] \[P(B)=\frac{1}{3}\] and \[P(A\cup B)=\frac{2}{3}.\] Are the events A and B mutually independent?

Answer:

Given,\[P\,(A)=\frac{1}{2},\]\[P\,(B)=\frac{1}{3}\]and \[P\,(A\cup B)=\frac{2}{3}\] \[\therefore \]      \[P\,(A\cap B)=P\,(A)+P\,(B)-P\,(A\cup B)\] \[=\frac{1}{2}+\frac{1}{3}-\frac{2}{3}=\frac{3+2-4}{6}=\frac{1}{6}\] Here, \[P\,(A)\cdot P\,(B)=\frac{1}{2}\times \frac{1}{3}=\frac{1}{6}\] \[\therefore \]\[P\,(A)\cdot P\,(B)=P\,(A\cap B)\] \[\therefore \]Events A and B are mutually independent.