If A and B are independent events, then which of the following statements are TRUE?

(A) $P(A \cap B) = P(A).P(B)$

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

(C) $P(A \cup B) = P(A) + P(B) - P(A).P(B)$

(D) $P(A \cap B) = P(A). P(B|A)$

Choose the correct answer from the options given below:

For two independent events A and B, one event happening doesn't affect the probability of the other.

The key formula for independent events:

$P(A \cap B) = P(A) \times P(B)$

---

Statement (A): $P(A \cap B) = P(A).P(B)$

This is the definition of independent events.

Statement (A) is TRUE.

---

Statement (B): $P(A \cap B) = P(A) - P(B)$

If $P(A) = 0.3$ and $P(B) = 0.5$:

$P(A \cap B) = 0.3 - 0.5 = -0.2$

Probability cannot be negative.

Statement (B) is FALSE.

---

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

The general union formula:

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

For independent events, $P(A \cap B) = P(A) \times P(B)$

Substituting:

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

Statement (C) is TRUE.

---

Statement (D): $P(A \cap B) = P(A). P(B|A)$

The general multiplication formula (always valid):

$P(A \cap B) = P(A) \times P(B|A)$

For independent events, $P(B|A) = P(B)$ since A doesn't affect B.

Therefore:

$P(A \cap B) = P(A) \times P(B|A)$

$P(A \cap B) = P(A) \times P(B)$

Statement (D) is TRUE.

---

Statements (A), (C), and (D) are TRUE.