Question

If two events $A$ and $B$ such that $P\left(A^c\right)=0.3,P\left(B\right)=0.4$ and $P(A\cap B^c)=0.5$, $P[B/A\cup B^c)]=\frac{1}{k}$,then value of k is

Answer 1

$P(A\cup B^c)=P\left(A\right)+P\left(B^c\right)-P(A\cap B^c)=1-P\left(A^c\right)+P\left(B^c\right)-P(A\cap B^c)=0.7+0.6-0.5=0.8$
Therefore,
$P(B|A\cup B^c)=\frac{P(B\cap (A\cup B^c))}{P(A\cup B^c)}=\frac{P[(B\cap A)\cup (B\cap B^c)]}{P(A\cup B^c)}=\frac{P(A\cap B)}{P(A\cup B^c)}=\frac{P\left(A\right)-P(A\cap B^c)}{P(A\cup B^c)}=\frac{0.7-0.5}{0.8}=0.25$