The probabilities of solving a problem correctly by `A` and `B` are `(1)/(8)` and `(1)/(12)` respectively. Given that they obtain the same answer after solving a problem and the probability of a common mistake by them is `(1)/(1001)`, then probability that their solution is correct is (Assuming that if they commit different mistake, then their answers will differ)

Options

A. `(77)/(96)`
B. `(14)/(15)`
C. `(2)/(5)`
D. `(13)/(14)`

Explanation

To solve the problem, we need to find the probability that their solution is correct given that they obtained the same answer. We will break this down step by step.

### Step 1: Define the probabilities
Let:
- \( P(A) = \frac{1}{8} \) (Probability that A solves the problem correctly)
- \( P(B) = \frac{1}{12} \) (Probability that B solves the problem correctly)

The probabilities of them solving incorrectly are:
- \( P(A') = 1 - P(A) = 1 - \frac{1}{8} = \frac{7}{8} \)
- \( P(B') = 1 - P(B) = 1 - \frac{1}{12} = \frac{11}{12} \)

### Step 2: Calculate the probability that both solve correctly
The probability that both A and B solve the problem correctly is given by:
\[ P(A \cap B) = P(A) \times P(B) = \frac{1}{8} \times \frac{1}{12} = \frac{1}{96} \]

### Step 3: Calculate the probability that both solve incorrectly
The probability that both A and B solve the problem incorrectly is given by:
\[ P(A' \cap B') = P(A') \times P(B') = \frac{7}{8} \times \frac{11}{12} = \frac{77}{96} \]
However, since they can only have the same answer if they make a common mistake, we need to multiply this by the probability of a common mistake:
\[ P(A' \cap B' \text{ and common mistake}) = P(A' \cap B') \times P(\text{common mistake}) = \frac{77}{96} \times \frac{1}{1001} = \frac{77}{96096} \]

### Step 4: Calculate the total probability of obtaining the same answer
The total probability that they obtain the same answer is the sum of the probabilities of both solving correctly and both making a common mistake:
\[ P(\text{same answer}) = P(A \cap B) + P(A' \cap B' \text{ and common mistake}) = \frac{1}{96} + \frac{77}{96096} \]

To add these fractions, we need a common denominator. The least common multiple of 96 and 96096 is 96096. Therefore:
\[ P(A \cap B) = \frac{1}{96} = \frac{1001}{96096} \]
Now we can add:
\[ P(\text{same answer}) = \frac{1001}{96096} + \frac{77}{96096} = \frac{1078}{96096} \]

### Step 5: Calculate the probability that their solution is correct given they obtained the same answer
Using Bayes' theorem, we find the probability that their solution is correct given they obtained the same answer:
\[ P(\text{correct} | \text{same answer}) = \frac{P(A \cap B)}{P(\text{same answer})} = \frac{\frac{1}{96}}{\frac{1078}{96096}} = \frac{1001}{1078} \]

### Step 6: Simplify the fraction
To simplify \( \frac{1001}{1078} \):
- The GCD of 1001 and 1078 is 1, so the fraction is already in its simplest form.

Thus, the final answer is:
\[ \text{Probability that their solution is correct} = \frac{1001}{1078} \]