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If A and B are two independent events in a given sample space and the probability that both A and B occur is 0.16 while the probability that neither occurs is 0.36 the probabilities P[A) and P[B) are respectively
- a)0.6,0.6
- b)0.8,0.2
- c)0.4,0.4
- d)0.9,0.4
Correct answer is option 'C'. Can you explain this answer?
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Given Information:
- P(A ∩ B) = 0.16
- P(A' ∩ B') = 0.36
Calculating P(A) and P(B):
- P(A' ∩ B') = P((A ∪ B)') = 1 - P(A ∪ B)
- P(A' ∩ B') = 1 - (P(A) + P(B) - P(A ∩ B))
- 0.36 = 1 - (P(A) + P(B) - 0.16)
- 0.36 = 1 - P(A) - P(B) + 0.16
- P(A) + P(B) = 0.8
Solving for P(A) and P(B):
- From the above equation, we have P(A) + P(B) = 0.8
- Given that A and B are independent, P(A ∩ B) = P(A) * P(B)
- 0.16 = P(A) * P(B)
Substitute P(B) from the equation P(A) + P(B) = 0.8:
- 0.16 = P(A) * (0.8 - P(A))
- 0.16 = 0.8P(A) - P(A)^2
Solving the quadratic equation:
- P(A)^2 - 0.8P(A) + 0.16 = 0
- (P(A) - 0.4)^2 = 0
- P(A) = 0.4, P(B) = 0.4
Therefore, the probabilities P(A) and P(B) are both 0.4, which corresponds to option 'C'.
- P(A ∩ B) = 0.16
- P(A' ∩ B') = 0.36
Calculating P(A) and P(B):
- P(A' ∩ B') = P((A ∪ B)') = 1 - P(A ∪ B)
- P(A' ∩ B') = 1 - (P(A) + P(B) - P(A ∩ B))
- 0.36 = 1 - (P(A) + P(B) - 0.16)
- 0.36 = 1 - P(A) - P(B) + 0.16
- P(A) + P(B) = 0.8
Solving for P(A) and P(B):
- From the above equation, we have P(A) + P(B) = 0.8
- Given that A and B are independent, P(A ∩ B) = P(A) * P(B)
- 0.16 = P(A) * P(B)
Substitute P(B) from the equation P(A) + P(B) = 0.8:
- 0.16 = P(A) * (0.8 - P(A))
- 0.16 = 0.8P(A) - P(A)^2
Solving the quadratic equation:
- P(A)^2 - 0.8P(A) + 0.16 = 0
- (P(A) - 0.4)^2 = 0
- P(A) = 0.4, P(B) = 0.4
Therefore, the probabilities P(A) and P(B) are both 0.4, which corresponds to option 'C'.
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If A and B are two independent events in a given sample space and the probability that both A and B occur is 0.16 while the probability that neither occurs is 0.36 the probabilities P[A) and P[B) are respectivelya)0.6,0.6b)0.8,0.2c)0.4,0.4d)0.9,0.4Correct answer is option 'C'. Can you explain this answer?
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Solutions for If A and B are two independent events in a given sample space and the probability that both A and B occur is 0.16 while the probability that neither occurs is 0.36 the probabilities P[A) and P[B) are respectivelya)0.6,0.6b)0.8,0.2c)0.4,0.4d)0.9,0.4Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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