Let A and B be two events. Then which of the following statements are TRUE?

(A) $P(B|A) = \frac{P(A \cap B)}{P(A)}$, provided $P(A) \neq 0$

(B) $P(B') = 1 + P(B)$

(C) $P(A \cup B) = P(A) + P(B) + P(A \cap B)$

(D) $P(A \cap B) = P(A).P(B)$ If A and B are independent events

Choose the correct answer from the options given below:

Statement (A): $P(B|A) = \frac{P(A \cap B)}{P(A)}$, provided $P(A) \neq 0$

This is the standard formula for conditional probability. The probability of event B given that event A has occurred equals the probability of both A and B occurring divided by the probability of A.

Statement (A) is TRUE.

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Statement (B): $P(B') = 1 + P(B)$

The correct complement rule is:

$P(B') = 1 - P(B)$

The probability of the complement of an event equals 1 minus the probability of the event. The given formula is incorrect as it would give probabilities greater than 1.

Statement (B) is FALSE.

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Statement (C): $P(A \cup B) = P(A) + P(B) + P(A \cap B)$

The correct addition rule is:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

When calculating the probability of the union, we add $P(A)$ and $P(B)$, but the intersection $P(A \cap B)$ is counted twice, so it must be subtracted once.

Statement (C) is FALSE.

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Statement (D): $P(A \cap B) = P(A) \cdot P(B)$ if A and B are independent events

This is the definition of independent events. Two events are independent when the occurrence of one does not affect the probability of the other, and their joint probability equals the product of their individual probabilities.

Statement (D) is TRUE.

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Therefore, statements (A) and (D) are TRUE.