Question

A person is known to speak false once out of 4 times. If that person picks a card at random from a pack of 52 cards and reports that it is a king, then the probability that it is actually a king is

• A. \( \frac{1}{37} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{12}{37} \)
• D. \( \frac{25}{37} \)

Hint:

Bayes' Theorem is a powerful tool for finding conditional probabilities when the reverse condition is more easily computed. - Remember: \( P(A|B) = \frac{P(B|A)P(A)}{P(B)} \) - Carefully account for the different probabilities when someone speaks truth or lies.

Answer 1

The Correct Option is B

Step 1: Define events: - Let \( A \) denote the event that the card is a king. - Let \( B \) denote the event that the person reports the card is a king.
Step 2: Calculate \( P(A) \) and \( P(B|A) \): - The probability that the card is a king is: \[ P(A) = \frac{4}{52} = \frac{1}{13}. \] - The probability that the person reports the card as a king, given that it is a king (the person speaks truth): \[ P(B|A) = \frac{3}{4}. \] Step 3: Calculate \( P(B|A^c) \): - The probability that the person reports a king when it is not a king (the person speaks falsely): \[ P(B|A^c) = \frac{1}{4}. \] Step 4: Use the Law of Total Probability to find \( P(B) \): - The total probability that the person reports a king is: \[ P(B) = P(B|A)P(A) + P(B|A^c)(1 - P(A)) = \frac{3}{4} \times \frac{1}{13} + \frac{1}{4} \times \frac{12}{13}. \] Simplifying: \[ P(B) = \frac{3}{52} + \frac{12}{52} = \frac{15}{52}. \] Step 5: Apply Bayes' Theorem to find \( P(A|B) \), the probability that the card is actually a king given that the person reported it as a king: \[ P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{\frac{3}{4} \times \frac{1}{13}}{\frac{15}{52}} = \frac{3}{5}. \]