A purse contains 2 six-sided dice. One is normal fair die, while the other has two 1's, and two 5's. A die is picked up and rolled. Because of some secret magnetic attraction of the unfair die, there is `75%` chance of picking the unfair die and a `25%` chance of picking a fair die. The die is rolled and shows up the face 3. The probability that a fair die was picked up is

Options

A. `1//7`
B. `1//4`
C. `1//6`
D. `1//24`

Explanation

To solve the problem, we need to find the probability that a fair die was picked given that the die rolled a 3. We can use Bayes' theorem for this purpose.

### Step-by-Step Solution:

1. **Define the Events**:
- Let \( N \) be the event that the die picked is a normal (fair) die.
- Let \( M \) be the event that the die picked is the magnetic (unfair) die.
- Let \( A \) be the event that the die shows a 3.

2. **Assign Probabilities**:
- The probability of picking the normal die:
\[ P(N) = 0.25 = \frac{1}{4} \]
- The probability of picking the magnetic die:
\[ P(M) = 0.75 = \frac{3}{4} \]

3. **Calculate Conditional Probabilities**:
- The probability of rolling a 3 with the normal die:
\[ P(A|N) = \frac{1}{6} \]
- The probability of rolling a 3 with the magnetic die (which has two 3's):
\[ P(A|M) = \frac{2}{6} = \frac{1}{3} \]

4. **Calculate Total Probability of Rolling a 3**:
Using the law of total probability:
\[ P(A) = P(A|N) \cdot P(N) + P(A|M) \cdot P(M) \]
Substituting the values:
\[ P(A) = \left(\frac{1}{6} \cdot \frac{1}{4}\right) + \left(\frac{1}{3} \cdot \frac{3}{4}\right) \]
\[ P(A) = \frac{1}{24} + \frac{3}{12} \]
Converting \(\frac{3}{12}\) to \(\frac{6}{24}\):
\[ P(A) = \frac{1}{24} + \frac{6}{24} = \frac{7}{24} \]

5. **Apply Bayes' Theorem**:
We want to find \( P(N|A) \):
\[ P(N|A) = \frac{P(A|N) \cdot P(N)}{P(A)} \]
Substituting the known values:
\[ P(N|A) = \frac{\left(\frac{1}{6}\right) \cdot \left(\frac{1}{4}\right)}{\frac{7}{24}} \]
\[ P(N|A) = \frac{\frac{1}{24}}{\frac{7}{24}} = \frac{1}{7} \]

### Final Answer:
The probability that a fair die was picked given that the die rolled a 3 is:
\[ \boxed{\frac{1}{7}} \]