The probabilities of two events, A and B, are 0.40 and 0.30 respectively. The…
2024
The probabilities of two events, A and B, are 0.40 and 0.30 respectively. The probability that both A and B occur is 0.15. What is the probability that neither A nor B occurs?
- A.
0.40
- B.
0.65
- C.
0.45
- D.
0.35
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Show answer & explanation
Correct answer: C
Concept: For any two events A and B, the probability that at least one of them occurs (their union) follows the inclusion-exclusion rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Since “neither A nor B occurs” is the complement of “A or B occurs,” P(neither) = 1 − P(A ∪ B).
Applying this to the given values:
Identify the given values: P(A) = 0.40, P(B) = 0.30, P(A ∩ B) = 0.15.
Apply the union rule: P(A ∪ B) = 0.40 + 0.30 − 0.15 = 0.55.
Apply the complement rule: P(neither A nor B) = 1 − P(A ∪ B) = 1 − 0.55 = 0.45.
Cross-check: Partition the sample space into the four mutually exclusive Venn-diagram regions — only A (P(A) − P(A∩B) = 0.25), only B (P(B) − P(A∩B) = 0.15), both A and B (0.15), and neither (call it p). These must sum to 1: 0.25 + 0.15 + 0.15 + p = 1, giving p = 0.45 — the same result.