True / False: If two events are mutually exclusive, they must also be independent.

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Question restatement: True / False: If two events are mutually exclusive, they must also be independent.

Option 1: True. This statement is incorrect. Mutually exclusive events cannot occur at the same time; their joint probability P(A ∩ B) is 0. Independence requires that P(A ∩ B) = P(A)P(B). For two events that are mutually exclusive, P(A ∩ B) = 0, and this equality would only hold if at least one of P(A) or P(B) were 0. In most practical situations both events have positive probability, so 0 ≠ P(A)P(B), breaking independence. Therefore, mutual exclusivity does not imply independence.

Option 2: False. This statement correctly reflects the relationship between mutual exclusivity and independence. Since mutual exclusivity forces P(A ∩ B) = 0 while independence would require P(A ∩ B) = P(A)P(B), the two conditions cannot simultaneously hold unless one event has probability zero. Thus, mutual exclusivity does not guarantee independence, and the two concepts describe different relationships between events.

Additional nuance: It's also worth noting the converse is not true either—independence does not imply mutual exclusivity, because two independent events can both occur (if both have positive probability), yielding a nonzero intersection. The key takeaway is that the two properties are distinct and typically incompatible when both events have positive probability.

STP420 - Probability and Statistics STP420 - Module 4 - Quiz STP420 - Probability and Statistics STP420 - Module 4 - Quiz

True / False: If two events are mutually exclusive, they must also be independent.

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