question_answer

Let A and B are two events such that P= 3/5 and P(B)= 2/3, then

Statement-1: \[\frac{4}{15}\le P(A\cap B)\le \frac{3}{5}.\]

Statement-2: \[\frac{2}{5}\le P\left( \frac{A}{B} \right)\le \frac{9}{10}\]

A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

C) Statement-1 is true, Statement-2 is false.

D) Statement-1 is false, Statement-2 is true.

Correct Answer: B

Solution :

\[\because \] \[(A\cup B)\ge P(A)+P(B)-1\] \[\therefore \]\[P(A\cap B)=\frac{3}{5}+\frac{2}{3}-1\Rightarrow P(A\cap B)\ge \frac{4}{15}\]?(i) \[\because \]\[P(A\cap B)\le P(A)\Rightarrow P(A\cap B)\le \frac{3}{5}\] From (i) and (ii), \[\frac{4}{15}\le P(A\cap B)\le \frac{3}{5}\]                         ...(iii) From (iii), \[\frac{4}{15P(B)}\le \frac{P(A\cap B)}{P(B)}\le \frac{3}{5P(B)}\]\[\Rightarrow \]\[\frac{2}{5}\le P\left( \frac{A}{B} \right)\le \frac{9}{10}\]