What is the correct answer to the multiple-choice question about the relationship between event independence and mutual exclusivity?

Conditions

• A and B are any two events in a random experiment.

Reasoning, step by step

1. Analyze option C: Mutually exclusive events with nonzero probabilities are not independent, so C is false.
2. Analyze option D: An impossible event is mutually exclusive with and independent of any event, so D is false.
3. Analyze option A: A non-empty intersection does not guarantee independence (e.g., subset relationships), so A is false.
4. Analyze option B: A non-empty intersection allows for the possibility of independence, making B the correct answer.

Example

The video evaluates four options regarding events A and B. Options claiming mutual exclusivity implies necessary independence or necessary dependence are refuted by counterexamples. Option B, stating that a non-empty intersection means events may be independent, is confirmed as correct.

Common misconceptions

• Believing that mutually exclusive events are always dependent, ignoring the special case of impossible events.
• Assuming that events that can occur together must be independent.

Watch the explanation

Explore next

• Why, when AB ≠ ∅, can we only say the events 'might possibly' be independent—not 'definitely' independent?
• Are impossible events independent of any event?