Explanation
1. Define the Events:
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T: The statement is true.
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F: The statement is false.
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E: Both A and B agree on the statement.
2. Given Probabilities:
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P(A speaks truth)=x
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P(B speaks truth)=y
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P(A tells a lie)=1−x
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P(B tells a lie)=1−y
3. Determine the Probability of Agreement (E):
Two witnesses can agree in two scenarios:
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Case 1: Both speak the truth.
P(E∣T)=P(A truth)×P(B truth)=xy
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Case 2: Both tell a lie.
P(E∣F)=P(A lie)×P(B lie)=(1−x)(1−y)
4. Apply Bayes' Theorem:
We want to find P(T∣E), the probability the statement is true given they agree.
P(T∣E)=P(T)⋅P(E∣T)+P(F)⋅P(E∣F)P(T)⋅P(E∣T)
Assuming the prior probability of the statement being true or false is equal (P(T)=P(F)=21), the formula simplifies to:
Final Answer:
The probability that the statement is true is xy+(1−x)(1−y)xy. The correct option is (a).
