If A and B are independent events, then which of the following is not true?

Independent events occur when one event happening doesn't affect the other event's probability.

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

Mutually exclusive events cannot happen at the same time.

For mutually exclusive events: $P(A \cap B) = 0$

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Option 4 states that A and B are mutually exclusive. This is NOT true.

If A and B are both independent AND mutually exclusive:

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

$P(A \cap B) = 0$ (from mutual exclusivity)

Therefore: $P(A) \times P(B) = 0$

This means either $P(A) = 0$ or $P(B) = 0$, implying one event is impossible.

For typical events with non-zero probabilities, independent events are NOT mutually exclusive.

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Option 1: $A'$ and $B'$ are independent.

When A and B are independent, their complements are also independent.

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

This statement is true.

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Option 2: A and $B'$ are independent.

When A and B are independent, A and the complement of B are also independent.

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

This statement is true.

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Option 3: $A'$ and B are independent.

When A and B are independent, the complement of A and B are also independent.

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

This statement is true.

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Option 4 is the correct answer.

Independent events can occur together, while mutually exclusive events cannot. These are fundamentally different concepts.