QJEEM14Ma15ad: JEE Main 2014 Mathematics - Class 12 JEEM14Ma15ad Solution: Let and be two events such that and where stands for the…

$P(\overline {A \cup B} ) = {1 \over 6}$

or, $1 - P(A \cup B) = {1 \over 6}$

$\therefore$ $P(A \cup B) = 1 - {1 \over 6} = {5 \over 6};$ $P(A \cap B) = {1 \over 4}$ ; and $P(\overline A ) = {1 \over 4};$

$\therefore$ $P(A) = 1 - P(\overline A ) = 1 - {1 \over 4} = {3 \over 4}$

We know, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

or, ${5 \over 6} = {3 \over 4} + P(B) - {1 \over 4}$

or, $P(B) = {5 \over 6} - {1 \over 2} = {1 \over 3}$

Now, $P(A)\,.\,P(B) = {3 \over 4}.\,{1 \over 3} = {1 \over 4} = P(A \cap B)$

i.e., events A and B are mutually independent.

Since the probability of A and B are different, so they are not equally likely events.

Therefore, (A) is the correct option.