Final answer:
To find the probability that the answer was correct, we can use Bayes' theorem. Given the probabilities of A and B solving the problem correctly, along with the chances of a common mistake, we can calculate the probability that A's answer was correct given that they both obtained the same answer. Using Bayes' theorem, we find that the probability is 12/13.
Step-by-step explanation:
To find the probability that the answer was correct, we need to use Bayes' theorem. Let's define the events:
A: A solves the problem correctly
B: B solves the problem correctly
We are given:
P(A) = 1/8
P(B) = 1/12
P(common mistake) = 1/1001
We want to find P(A|same answer), which represents the probability that A's answer was correct given that they both obtained the same answer.
Using Bayes' theorem, we have:
P(A|same answer) = (P(same answer|A) * P(A)) / P(same answer)
The probability that the answer is the same, given A's answer is correct, can be calculated as:
P(same answer|A) = P(A and B both solve the problem correctly) = P(A) * P(B) = (1/8)(1/12)
The probability that the answer is the same can be calculated as:
P(same answer) = P(A and B both solve the problem correctly) + P(not same answer)
P(not same answer) = P(A solves the problem correctly and B makes a common mistake) + P(B solves the problem correctly and A makes a common mistake)
Since A and B's chances of solving the problem correctly are given, we can calculate P(not same answer):
P(not same answer) = (1/8)(1/1001) + (1/12)(1/1001)
Now we can substitute the values into the equation to find P(A|same answer):
P(A|same answer) = [(1/8)(1/12)] / [(1/8)(1/12) + (1/8)(1/1001) + (1/12)(1/1001)]
Simplifying the expression, we get:
P(A|same answer) = 12/13
Therefore, the probability that A's answer was correct given that they both obtained the same answer is 12/13.