Question

A card from a well-shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.

Hint:

When solving problems involving missing or conditional probabilities, use Bayes' Theorem and clearly define all events and conditional probabilities.

Answer 1

Step 1: Event Definition
Let \( E_1 \) denote the event that the lost card is a King. Let \( E_2 \) denote the event that the lost card is not a King. Let \( A \) denote the event of drawing a King from the remaining 51 cards.
Step 2: Probability Assignment
The probabilities of these events are assigned as follows: \[ P(E_1) = \frac{1}{13}, \quad P(E_2) = \frac{12}{13}, \quad P(A|E_1) = \frac{3}{51}, \quad P(A|E_2) = \frac{4}{51} \] 
Step 3: Application of Bayes' Theorem
The objective is to calculate \( P(E_1|A) \). Bayes' Theorem provides the formula: \[ P(E_1|A) = \frac{P(A|E_1) \cdot P(E_1)}{P(A|E_1) \cdot P(E_1) + P(A|E_2) \cdot P(E_2)} \] Upon substitution of the assigned probabilities: \[ P(E_1|A) = \frac{\frac{1}{13} \cdot \frac{3}{51}}{\frac{1}{13} \cdot \frac{3}{51} + \frac{12}{13} \cdot \frac{4}{51}} = \frac{\frac{3}{663}}{\frac{3}{663} + \frac{48}{663}} = \frac{3}{51} = \frac{1}{17} \] 
Step 4: Conclusion
The probability that the lost card was a King is \( \frac{1}{17} \).