Question

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

• A. $P(A / B) \geq \frac{P(A)+P(B)-1}{P(B)}, P(B) \neq 0$ 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 $A P(A / B)=\frac{P(A \cap B)}{P(B)}$
$\Rightarrow \text { T.P.T. } P(A \cap B) \geq P(A)+P(B)-1$
or T.P.T., $1 \geq P ( A )+ P ( B )- P ( A \cap B )$
or T.P.T., $1 \geq P ( A \cup B )$
which is true $\Rightarrow(\bar{A})$ is correct
(B) and (C) are obvious