Explanation
Step 1: Define the Events
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C1: Both A and B solve the problem correctly.
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C2: Both A and B solve the problem incorrectly but get the same wrong answer (common error).
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S: Both A and B obtain the same answer.
Step 2: Calculate Probabilities of Solving
Step 3: Calculate Conditional Probabilities
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Case 1: Both are correct (C1)
If both are correct, they must have the same answer.
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Case 2: Both are wrong with the same answer (C2)
The probability that both are wrong is 32×43=126=21.
The probability of making a common error given they are both wrong is 201.
P(C2)=32×43×201=2406=401
Step 4: Apply Bayes' Theorem
We want to find P(C1∣S), the probability they are correct given they have the same answer:
P(C1∣S)=P(C1)⋅P(S∣C1)+P(C2)⋅P(S∣C2)P(C1)⋅P(S∣C1)
Substitute the values:
P(C1∣S)=(121×1)+(401×1)121×1
Find a common denominator for the bottom (LCM of 12 and 40 is 120):
P(C1∣S)=12010+120312010
Final Answer:
The probability that their answer is correct is 1310.
