Probability of Drawing All Face Cards from a Deck with No Replacement

In this article, we will explore the probability of drawing three face cards (kings, queens, and jacks) from a standard deck of 52 cards without replacement. We will break down the problem into manageable parts and provide a detailed explanation of the steps involved to arrive at the solution.

Introduction to Face Cards and Deck Composition

A standard deck of playing cards contains 52 cards, divided into four suits: hearts, diamonds, clubs, and spades. Each suit includes 13 cards, which consist of the numbers 2 through 10, as well as the face cards (kings, queens, and jacks) and the ace. In total, there are 12 face cards in the deck, forming 4 each in the four suits.

Understanding the Problem

The question asks for the probability of drawing three face cards without replacement from a deck of 52 cards. This means that once a card is drawn, it is not put back into the deck.

Step-by-Step Solution

Step 1: Determine the Initial Probability

The probability of drawing a face card on the first draw is:

12/52 3/13

Step 2: Update the Deck After the First Draw

After the first face card is drawn, there are now 51 cards left in the deck, with 11 face cards remaining. The probability of drawing a face card on the second draw is:

11/51

Step 3: Continue to Update the Deck After the Second Draw

After the second face card is drawn, there are 50 cards left, with 10 face cards remaining. The probability of drawing a face card on the third draw is:

10/50 1/5

Step 4: Calculate the Overall Probability

To find the overall probability of drawing three face cards in succession without replacement, multiply the individual probabilities together:

(12/52) * (11/51) * (10/50)

First, simplify the fractions:

(3/13) * (11/51) * (1/5) 33/3315

Now, further simplify the fraction:

33/3315 22/425 ≈ 0.0518

Conclusion

The probability of drawing three face cards from a deck of 52 cards without replacement is 22/425 or approximately 0.0518. This means that there is roughly a 5.18% chance of drawing three face cards in a row without putting any of them back into the deck.

Key Points Recap:

Understand the total number of face cards in a deck (12). Calculate the probability for each draw, considering the updated deck size. Multiply the individual probabilities to get the overall probability.

By breaking down the problem step-by-step, you can easily understand and solve similar probability questions involving decks of cards.