Question

Let two events $A \& B$ are such that $P\left(A^c\right)=0.3, P(B)=0.4 \& P\left(A B^c\right)=0.5$. If $P\left[B /\left(A U B^c\right)\right]$ is expressed as $\frac{p}{q}$ in lowest rational. Find $(p+q)$.

Answer 1

$P\left(A^c\right)=0.3 \Rightarrow P(A)=0.7$
$P(B)=0.4 ; P\left(A \cap B^c\right)=0.5 $
$\text { or } P(A)-P(A \cap B)=0.5$
$\therefore P(A \cap B)=0.7-0.5=0.2$
$\text { Now } P\left(B / A \cup B^c\right)=\frac{P\left(B \cap\left(A \cup B^c\right)\right)}{P\left(A \cup B^c\right)}$
$=\frac{P(B \cap A)+P\left(B \cap B^c\right)}{P\left(A \cup B^c\right)}=\frac{P(A \cap B)}{P(A)+P\left(B^c\right)-P\left(A \cap B^c\right)} $
$=\frac{P(A \cap B)}{P(A)+1-P(B)-(P(A)-P(A \cap B))}=\frac{P(A \cap B)}{1-P(B)+P(A \cap B)}=\frac{0.2}{1-0.4+0.2} $
$=\frac{0.2}{0.8}=\frac{1}{4}=\frac{p}{q} \Rightarrow(p+q)=1+4=5$