If A and B are independent events, then which of the following is/are true?

(A) $\bar{A}$ and B are independent events

(B) $P(A \cap B) = 0$

(C) $\bar{A}$ and $\bar{B}$ are independent events

(D) $P(A \cap B) = P(A) + P(B)$

Choose the correct answer from the options given below:

When A and B are independent events:

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

This means one event happening does not affect the probability of the other.

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Statement (A): $\bar{A}$ and B are independent

$P(\bar{A} \cap B) = P(B) - P(A \cap B)$

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

$= P(B) \times [1 - P(A)]$

$= P(B) \times P(\bar{A})$

Therefore, $\bar{A}$ and B are independent.

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Statement (B): $P(A \cap B) = 0$

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

This equals 0 only if $P(A) = 0$ or $P(B) = 0$.

Example: Tossing a coin twice where A = first toss is heads and B = second toss is heads.

$P(A) = 0.5$, $P(B) = 0.5$

$P(A \cap B) = 0.5 \times 0.5 = 0.25 \neq 0$

Independent events can happen together, so this statement is false.

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Statement (C): $\bar{A}$ and $\bar{B}$ are independent

$P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B)$

$= 1 - [P(A) + P(B) - P(A \cap B)]$

$= 1 - P(A) - P(B) + P(A) \times P(B)$

$= [1 - P(A)] \times [1 - P(B)]$

$= P(\bar{A}) \times P(\bar{B})$

Therefore, $\bar{A}$ and $\bar{B}$ are independent.

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

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

The formula $P(A \cap B) = P(A) + P(B)$ is mathematically incorrect.

Note: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ is the formula for union, not intersection.

This statement is false.

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Statements (A) and (C) are true.

If A and B are independent, then all combinations remain independent: A and B, $\bar{A}$ and B, A and $\bar{B}$, and $\bar{A}$ and $\bar{B}$.