Question

If A and B are two events such that

$P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{1}{3}$ and $P(A\cap B)=\frac{1}{4},$ then find

$P\left(\frac{A^{\prime }}{B^{\prime }}\right)$

Answer 1

$P\left(\frac{A^{\prime }}{B^{\prime }}\right)=\frac{P(A^{\prime }\cap B^{\prime })}{P\left(B^{\prime }\right)}=\frac{1-P(A\cup B)}{1-P\left(B\right)}=\frac{1-\left[P\right(A)+P(B)-P(A\cap B\left)\right]}{1-P\left(B\right)}=\frac{1-[\frac{1}{2}+\frac{1}{3}-\frac{1}{4}]}{1-\frac{1}{3}}=\frac{1-(\frac{5}{6}-\frac{1}{4})}{\frac{2}{3}}=\frac{1-\frac{14}{24}}{\frac{2}{3}}=\frac{\frac{10}{24}}{\frac{2}{3}}=\frac{30}{48}=\frac{5}{8}$