Question

If $P(A/B)+P(\overline{A}/\overline{B})=1$ and $0<P(A)<1$; and $0<P(B)<1$ the which of the following does not hold good?

• A. A and B are equiprobable
• B. A and B are disjoint
• C. A and B are independent
• D. A and B are neither disjoint nor independent

Answer 1

Answer: (A), (B), (D)

$\frac{P(A\cap B)}{P(B)}+\frac{P(\overline{A}\cap \overline{B})}{P(\overline{B})}=1$
$\frac{P(A\cap B)}{P(B)}=1-\frac{1-P(A\cup B)}{1-P(B)}=\frac{1-P(B)-1+P(A\cup B)}{1-P(B)}$
$\frac{P(A\cap B)}{P(B)}=\frac{P(A\cup B)-P(B)}{1-P(B)}=\frac{P(A)-P(A\cap B)}{1-P(B)}$
$\therefore P(A\cap B)-P(A\cap B)P(B)=P(A)P(B)-P(B)P(A\cap B)$
$P(A\cap B)=P(A)P(B)\Rightarrow (C)$