Independent vs mutually exclusive events: which statement is correct?

Independent vs mutually exclusive events: which statement is correct?

• A. Independent: P(A ∩ B) = P(A)P(B); Mutually exclusive: P(A ∪ B) = P(A) + P(B) when disjoint.
• B. Independent events cannot occur at the same time.
• C. Mutually exclusive means P(A ∩ B) = 0 always.
• D. Independent events require P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Answer: (A)

Understanding the difference between independence and mutual exclusivity helps you pick the right rule quickly. For independent events, the chance both occur is the product of their separate chances: P(A ∩ B) = P(A)P(B). This is the defining feature of independence—the occurrence of one event does not change the likelihood of the other. For mutually exclusive (disjoint) events, they cannot happen at the same time. That means P(A ∩ B) = 0, and the probability that either occurs is simply the sum of their individual probabilities: P(A ∪ B) = P(A) + P(B) when they are disjoint. The statement that combines both ideas in one line—independence is P(A ∩ B) = P(A)P(B); mutual exclusivity means P(A ∪ B) = P(A) + P(B) when disjoint—captures the standard, direct rules for these two concepts, so it’s the best choice. Why the others don’t fit as cleanly: independent events can occur together, with probability P(A)P(B), so saying they cannot occur at the same time is false. Mutual exclusivity does imply P(A ∩ B) = 0, which is true, but the most practical takeaway is the union rule for disjoint events. The general inclusion–exclusion formula P(A ∪ B) = P(A) + P(B) − P(A ∩ B) holds for any events and does not distinguish independence from mutual exclusivity.

Understanding the difference between independence and mutual exclusivity helps you pick the right rule quickly. For independent events, the chance both occur is the product of their separate chances: P(A ∩ B) = P(A)P(B). This is the defining feature of independence—the occurrence of one event does not change the likelihood of the other.

For mutually exclusive (disjoint) events, they cannot happen at the same time. That means P(A ∩ B) = 0, and the probability that either occurs is simply the sum of their individual probabilities: P(A ∪ B) = P(A) + P(B) when they are disjoint.

The statement that combines both ideas in one line—independence is P(A ∩ B) = P(A)P(B); mutual exclusivity means P(A ∪ B) = P(A) + P(B) when disjoint—captures the standard, direct rules for these two concepts, so it’s the best choice.

Why the others don’t fit as cleanly: independent events can occur together, with probability P(A)P(B), so saying they cannot occur at the same time is false. Mutual exclusivity does imply P(A ∩ B) = 0, which is true, but the most practical takeaway is the union rule for disjoint events. The general inclusion–exclusion formula P(A ∪ B) = P(A) + P(B) − P(A ∩ B) holds for any events and does not distinguish independence from mutual exclusivity.