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:

$\frac{5}{12}$

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

$P(A' \cap B')=1-P(A\cup B)$

$=1-\frac{7}{12}$

$=\frac{5}{12}$

$P(A' \cap B')=\frac{5}{12}$.