For two events A and B, P(A) = 0.2; P(B) = 0.6, and P(A or B) = 0.8. Which of the following statements must be true?

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BlackTom AI · Accepted Answer

We start by noting the given probabilities: P(A) = 0.2, P(B) = 0.6, and P(A or B) = 0.8. Using the inclusion–exclusion principle, P(A or B) = P(A) + P(B) − P(A ∩ B).

Plugging in the values: 0.8 = 0.2 + 0.6 − P(A ∩ B) = 0.8 − P(A ∩ B).
This implies P(A ∩ B) = 0. That means A and B do not occur together; hence they are disjoint (mutually exclusive).

Next, to assess independence, recall that A and B are independent if and only if P(A ∩ B) = P(A)P(B). Here, P(A ∩ B) = 0, while P(A)P(B) = 0.2 × 0.6 = 0.12. Since 0 ≠ 0.12, the events are not independent.

With these conclusions, evaluate each option:
- Option 1: "A and B are independent, but not disjoint" would require P(A ∩ B) = P(A)P(B) and a nonzero intersection. We have P(A ∩ B) = 0 and P(A)P(B) = 0.12, so this is incorrect.
- Option 2: "A and B are independent and disjoint" would demand both independence and zero intersection. While P(A ∩ B) = 0, independence fails because P(A ∩ B) ≠ P(A)P(B). So this is false.
- Option 3: "A and B are neither disjoint nor independent" asserts a positive intersection and dependence. Since P(A ∩ B) = 0, they are disjoint, making this option false.
- Option 4: "A and B are disjoint, but not independent" matches our findings: disjoint because P(A ∩ B) = 0, and not independent since P(A ∩ B) ≠ P(A)P(B).

Thus, the correct interpretation is that A and B are disjoint but not independent, based on the given probabilities and the fundamental relationships between disjointness, independence, and intersection probability.

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For two events A and B, P(A) = 0.2; P(B) = 0.6, and P(A or B) = 0.8. Which of the following statements must be true?

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