Statistics And Probability Question 516
Question: If $ P(A)=2/3 $ , $ P(B)=1/2 $ and $ P(A\cup B)=5/6 $ then events A and B are
[Kerala (Engg.) 2002]
Options:
A) Mutually exclusive
B) Independent as well as mutually exhaustive
C) Independent
D) Dependent only on A
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Answer:
Correct Answer: C
Solution:
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First, recall the addition rule of probability: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
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We’re given: P(A) = 2/3 P(B) = 1/2 P(A ∪ B) = 5/6
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Let’s substitute these into the addition rule: 5/6 = 2/3 + 1/2 - P(A ∩ B)
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Simplify: 5/6 = 4/6 + 3/6 - P(A ∩ B) 5/6 = 7/6 - P(A ∩ B)
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Solve for P(A ∩ B): P(A ∩ B) = 7/6 - 5/6 = 1/3
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Now, let’s consider each option:
A) If A and B were mutually exclusive, P(A ∩ B) would be 0. It’s not, so they’re not mutually exclusive.
B) They’re not mutually exhaustive because P(A ∪ B) ≠ 1. They’re also not independent (we’ll see why in C).
C) For independence, we would need P(A ∩ B) = P(A) * P(B). P(A) * P(B) = 2/3 * 1/2 = 1/3 This happens to be true! So A and B are independent.
D) They’re not dependent only on A because they’re independent.
Therefore, the correct answer is C) Independent.
BETA
