For any two events A and B in a sample space:

Engineering

Mathematics

Basic Definitions Related to Probability

Addition Theorem on Probability

Conditional Probability

Question

For any two events A and B in a sample space:

$P (A / B) \geq \frac{P (A) P (B) - 1}{P (B)} P (B) \neq 0$ is always true

$P(A⋂) = P(A) - P(A⋂B)$ , does not hold

$P (A \cup B) = 1 - P (\bar{A}) P (\bar{B})$ , if A and B are independent

$P (A \cup B) = 1 - P (\bar{A}) P (\bar{B})$ , if A and B are disjoint.

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(a): Option (a) is correct as LHS is always non negative while the RHS is always non-positive {since P(A) x P(B) is always less than or equal to 1 since each is less than or equal to 1}.
(b): LHS represents the region defined by A but not B or only A which is the same as RHS
(c): LHS refers to the union of A and B.
RHS = 1 – P(A'⋂B') = 1 – P(AUB') = P(A⋂B)
(d): It is obviously false.