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

Engineering

Mathematics

Conditional Probability

Question

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

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

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

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

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We know that

P (\frac{B}{A}) = \frac{P (B)}{P (A \cap B)}

= \frac{P (B)}{P (A) + P (B) - P (A \cup B)}

Since, P(A∪B) < 1 therefore

− P(A∪B) > − 1

⇒ P(A) + P(B)− + (A∪B) > P(A) + P(B) − 1

\Rightarrow \frac{P (A) + P (B) - P (A \cup B)}{P (B)} > \frac{P (A) + P (B) - 1}{P (B)}

\Rightarrow P (\frac{A}{B}) > \frac{P (A) + P (B) - 1}{P (B)}

Thus option (A) is correct and (C) is also correct.

Since, P(A∪B) = P(A) + P(B) − P(A∩B)

P(A∪B) = P(A) + P(B) − P(A)P(B)

If A and B are independent

= 1[1 − P(A)][1 − P(B)]

= 1 - P (A) P (B)

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