Question
In an experiment, the probability that event A occurs is 3/5, the probability that event B occurs is 1/3, and the probability that events A and B both occur is 2/9.
Answers
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GPT-5 mini
We can compute several useful probabilities.
Given P(A)=3/5, P(B)=1/3, P(A∩B)=2/9:
- P(A ∪ B) = P(A)+P(B)−P(A∩B) = 3/5 + 1/3 − 2/9 = 32/45.
- P(neither A nor B) = 1 − P(A ∪ B) = 13/45.
- P(A | B) = P(A∩B)/P(B) = (2/9)/(1/3) = 2/3.
- P(B | A) = P(A∩B)/P(A) = (2/9)/(3/5) = 10/27.
Also, since P(A)P(B) = (3/5)(1/3) = 1/5 ≠ 2/9 = P(A∩B), A and B are not independent. They are not mutually exclusive either because P(A∩B) > 0.
Given P(A)=3/5, P(B)=1/3, P(A∩B)=2/9:
- P(A ∪ B) = P(A)+P(B)−P(A∩B) = 3/5 + 1/3 − 2/9 = 32/45.
- P(neither A nor B) = 1 − P(A ∪ B) = 13/45.
- P(A | B) = P(A∩B)/P(B) = (2/9)/(1/3) = 2/3.
- P(B | A) = P(A∩B)/P(A) = (2/9)/(3/5) = 10/27.
Also, since P(A)P(B) = (3/5)(1/3) = 1/5 ≠ 2/9 = P(A∩B), A and B are not independent. They are not mutually exclusive either because P(A∩B) > 0.