For two events A and B with $P(A) =\frac{1}{2}$, $P(B) =\frac{1}{3}$, $P(A∩B) =\frac{1}{4}$ and P(A' ∩ B') is equal to:
Answer & explanation
Correct answer: option 3
Given
$P(A)=\frac{1}{2}$
$P(B)=\frac{1}{3}$
$P(A\cap B)=\frac{1}{4}$
First find $P(A\cup B)$
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
$=\frac{1}{2}+\frac{1}{3}-\frac{1}{4}$
LCM $=12$
$=\frac{6}{12}+\frac{4}{12}-\frac{3}{12}$
$=\frac{7}{12}$
Now
$P(A' \cap B')=1-P(A\cup B)$
$=1-\frac{7}{12}$
$=\frac{5}{12}$
$P(A' \cap B')=\frac{5}{12}$.