Question

From a well-shuffled pack of 52 cards, three cards were drawn one after another without any replacement. What is the probability that the first two cards be King and the third be Ace?

Hint:

When cards are drawn without replacement, reduce both numerator and denominator step by step. Always multiply successive probabilities for ordered outcomes.

Answer 1

We have a total of 52 cards. - Number of Kings in the deck = 4 - Number of Aces in the deck = 4

Step 1: Probability that the first card is King. \[ P(\text{1st is King}) = \frac{4}{52} = \frac{1}{13} \]

Step 2: Probability that the second card is also King. After one King is drawn, 3 Kings remain and 51 cards are left. \[ P(\text{2nd is King}) = \frac{3}{51} = \frac{1}{17} \]

Step 3: Probability that the third card is Ace. Now, 50 cards remain (4 Aces still there). \[ P(\text{3rd is Ace}) = \frac{4}{50} = \frac{2}{25} \]

Step 4: Multiply probabilities (independent successive events). \[ P(\text{First two Kings and third Ace}) = \frac{1}{13} \times \frac{1}{17} \times \frac{2}{25} \] \[ = \frac{2}{5525} \]

Final Answer: \[ \boxed{\dfrac{2}{5525}} \]