A and B are two students. Their probabilities of solving a problem correctly are 1/4 and 1/5 respectively. If the probability of their making a common error is 1/40, and they obtain the same answer, then the probability of their answer is correct is

Options

A. `1//12`
B. `1//20`
C. `10//13`
D. `13//200`

Answer

To solve the problem, we need to find the probability that A and B's answer is correct given that they obtained the same answer. We will use Bayes' theorem for this calculation.

Let:
- \( P(C) \) = Probability that their answer is correct
- \( P(S) \) = Probability that they get the same answer
- \( P(S|C) \) = Probability that they get the same answer given that their answer is correct
- \( P(S|C') \) = Probability that they get the same answer given that their answer is incorrect

### Step 1: Calculate the probabilities of A and B solving the problem correctly and incorrectly.
- Probability that A solves the problem correctly: \( P(A) = \frac{1}{4} \)
- Probability that A does not solve the problem correctly: \( P(A') = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4} \)

- Probability that B solves the problem correctly: \( P(B) = \frac{1}{5} \)
- Probability that B does not solve the problem correctly: \( P(B') = 1 - P(B) = 1 - \frac{1}{5} = \frac{4}{5} \)

### Step 2: Calculate the probability of them making a common mistake.
- Probability of making a common mistake: \( P(M) = \frac{1}{40} \)

### Step 3: Calculate \( P(S|C) \).
If both A and B are correct, they will definitely get the same answer:
- \( P(S|C) = 1 \)

### Step 4: Calculate \( P(S|C') \).
If both A and B are incorrect, they will get the same answer if they make a common mistake:
- \( P(S|C') = P(M) = \frac{1}{40} \)

### Step 5: Calculate \( P(S) \).
Using the law of total probability:
\[ P(S) = P(S|C) \cdot P(C) + P(S|C') \cdot P(C') \]
Where:
- \( P(C) = P(A) \cdot P(B) = \frac{1}{4} \cdot \frac{1}{5} = \frac{1}{20} \)
- \( P(C') = 1 - P(C) = 1 - \frac{1}{20} = \frac{19}{20} \)

Now substituting the values:
\[ P(S) = 1 \cdot \frac{1}{20} + \frac{1}{40} \cdot \frac{19}{20} \]
Calculating the second term:
\[ \frac{1}{40} \cdot \frac{19}{20} = \frac{19}{800} \]
So:
\[ P(S) = \frac{1}{20} + \frac{19}{800} \]
Converting \( \frac{1}{20} \) to have a common denominator:
\[ \frac{1}{20} = \frac{40}{800} \]
Thus:
\[ P(S) = \frac{40}{800} + \frac{19}{800} = \frac{59}{800} \]

### Step 6: Calculate \( P(C|S) \) using Bayes' theorem.
\[ P(C|S) = \frac{P(S|C) \cdot P(C)}{P(S)} \]
Substituting the values we have:
\[ P(C|S) = \frac{1 \cdot \frac{1}{20}}{\frac{59}{800}} = \frac{\frac{1}{20}}{\frac{59}{800}} = \frac{800}{20 \cdot 59} = \frac{40}{59} \]

Thus, the probability that their answer is correct given that they obtained the same answer is:
\[ \boxed{\frac{40}{59}} \]