True or False. All independent events are disjoint.

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Question: True or False. All independent events are disjoint.

Start by recalling the meanings of the two concepts. Independent events satisfy P(A ∩ B) = P(A) · P(B); the occurrence of one event does not affect the probability of the other. Disjoint (mutually exclusive) events satisfy A ∩ B = ∅, meaning both events cannot happen at the same time.

Now evaluate the statement. If two events are independent and both have positive probability, then P(A ∩ B) = P(A) · P(B) > 0, which means A and B can occur together. This violates the definition of disjoint events, which cannot occur together. Therefore, independence does not imply disjointness.

A simple example helps: flip a fair coin twice. Let A be “heads on the first flip” and B be “heads on the second flip.” These events are independent because the outcome of the first flip does not affect the second. However, they are not disjoint because it is perfectly possible for both to occur (heads on both flips), so A ∩ B ≠ ∅. Conversely, disjoint events cannot both happen, so if A and B were disjoint, P(A ∩ B) would be 0, which would contradict independence when P(A) and P(B) are positive.

Consider edge cases briefly: if one of the events has probability zero, independence can hold with A ∩ B empty, but the general statement that all independent events are disjoint is still not true in the common, nontrivial case where both events can occur with positive probability.

In summary, independence describes a lack of influence between events, while disjointness forbids simultaneous occurrence. The two concepts are distinct, and independence does not entail disjointness in general.

STAT 1450.02 SP2025 (28132) Checkpoint 13.1 & 13.2

True or False. All independent events are disjoint.

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