A man is known to speak truth 3 out of 4 times. He takes out a card at random from a well shuffled pack of 52 playing cards, and reports it is a king. The probability that its actually a king is

Options

A. `1/4`
B. `3/4`
C. `4/5`
D. `1/5`

Answer

To solve the problem, we need to find the probability that the card drawn is actually a king given that the man reports it as a king. We will use Bayes' theorem to calculate this conditional probability.

### Step-by-Step Solution:

1. **Define Events**:
- Let \( A \) be the event that the card drawn is actually a king.
- Let \( B \) be the event that the man reports the card as a king.

2. **Given Probabilities**:
- The probability that the man speaks the truth, \( P(T) = \frac{3}{4} \).
- The probability that the man speaks falsely, \( P(F) = 1 - P(T) = \frac{1}{4} \).
- The probability of drawing a king from a deck of 52 cards, \( P(A) = \frac{4}{52} = \frac{1}{13} \).
- The probability of not drawing a king, \( P(A') = 1 - P(A) = \frac{12}{13} \).

3. **Calculate \( P(B|A) \)**:
- If the card is actually a king, the probability that he reports it as a king (truth) is \( P(B|A) = P(T) = \frac{3}{4} \).

4. **Calculate \( P(B|A') \)**:
- If the card is not a king, the probability that he reports it as a king (lie) is \( P(B|A') = P(F) = \frac{1}{4} \).

5. **Apply Bayes' Theorem**:
- We want to find \( P(A|B) \):
\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]
- To find \( P(B) \), we use the law of total probability:
\[ P(B) = P(B|A) \cdot P(A) + P(B|A') \cdot P(A') \]
- Substitute the values:
\[ P(B) = \left(\frac{3}{4} \cdot \frac{1}{13}\right) + \left(\frac{1}{4} \cdot \frac{12}{13}\right) \]
\[ P(B) = \frac{3}{52} + \frac{12}{52} = \frac{15}{52} \]

6. **Final Calculation**:
- Now substitute \( P(B) \) back into Bayes' theorem:
\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{\left(\frac{3}{4} \cdot \frac{1}{13}\right)}{\frac{15}{52}} \]
\[ P(A|B) = \frac{\frac{3}{52}}{\frac{15}{52}} = \frac{3}{15} = \frac{1}{5} \]

### Conclusion:
The probability that the card is actually a king given that it is reported as a king is \( \frac{1}{5} \).