Question

For any two events A \& B defined on a sample space,

• A. $P(A/B)\ge \frac{P(A)+P(B)-1}{P(B)},P(B)\ne 0$ is always true
• B. $P(A\cap \bar{B})=P(A)-P(A\cap B)$
• C. $P(A\cup B)=1-P\left(A^c\right)\cdot P\left(B^c\right)$, if $A\&B$ are independent
• D. $P(A\cup B)=1-P\left(A^c\right)\cdot P\left(B^C\right)$, if $A\&B$ are disjoint

Answer 1

Answer: (A), (B), (C)

For $AP(A/B)=\frac{P(A\cap B)}{P(B)}$
$\Rightarrow \text{ T.P.T. }P(A\cap B)\ge P(A)+P(B)-1$
or T.P.T., $1\ge P(A)+P(B)-P(A\cap B)$
or T.P.T., $1\ge P(A\cup B)$
which is true $\Rightarrow (\bar{A})$ is correct
(B) and (C) are obvious