Question

If two events $A$ and $B$ such that $P\left(A^c\right)$ $P\left(B\right)=0.4$ and $P\left(AB{}^c\right)$$=0.5,$ then $P\left[B/\right(A\cup B{}^c\left)\right]$=......

• A. $0.25$
• B. $0.50$
• C. $0.75$
• D. None of these
  

Answer 1

The correct option is A $0.25$

$P\left(A^c\right)=0.3\Rightarrow P\left(A\right)=0.7P\left(B\right)=0.4\Rightarrow P\left(B^c\right)=0.6$
And $P(A\cap B^c)=0.5$
Now, $P(A\cup B^c)=P\left(A\right)+P\left(B^c\right)-P(A\cap B^c)=0.7+0.6-0.5=0.8$
$\therefore P\left(\frac{B}{(A\cup B^c)}\right)=\frac{P\{B\cap (A\cup B^c)\}}{P(A\cup B^c)}=\frac{P\{(B\cap A)\cup (B\cap A^c)\}}{0.8}$
$=\frac{P\{(B\cap A)\cup \varphi \}}{0.8}=\frac{P(B\cap A)}{0.8}=\frac{0.7-0.5}{0.8}=\frac{0.2}{0.8}=\frac{1}{4}$