Probability of Drawing at Least One King from a Standard Deck

In this article, we will delve into the fascinating world of probability as applied to a common card game scenario. We will calculate the probability of drawing at least one king from a standard deck of 52 cards without replacement.

Introduction to Probability in Card Games

Card games have always been a source of entertainment and intellectual challenge. One such scenario involves drawing cards from a standard deck. A full deck consists of 52 cards, divided evenly into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, including one king. The question we will explore is the probability of drawing at least one king from three randomly selected cards from this deck without replacement.

Calculating the Probability Without Using Kings

Step 1: Total number of ways to choose 3 cards from 52

Let's start by calculating the total number of ways to choose 3 cards from a deck of 52 cards. This can be done using the combination formula, which is given by:

C(n, k) n! / [k!(n - k)!]

Here, n 52 (total number of cards) and k 3 (number of cards to be chosen).

C(52, 3) 52! / [3!(52 - 3)!] 52! / [3! * 49!] 52 * 51 * 50 / (3 * 2 * 1) 22100

This gives us 22,100 possible ways to draw 3 cards from a standard deck without replacement.

Step 2: Total number of ways to choose 3 cards from 48

Next, we need to calculate the number of ways to choose 3 cards from the remaining 48 cards (after removing all 4 kings).

C(48, 3) 48! / [3!(48 - 3)!] 48! / [3! * 45!] 48 * 47 * 46 / (3 * 2 * 1) 17296

This gives us 17,296 possible ways to draw 3 cards from a deck of 48 cards.

Step 3: Calculating the Probability of Drawing No Kings

The probability of drawing no kings is calculated by dividing the number of ways to choose 3 cards from 48 (without any kings) by the total number of ways to choose 3 cards from 52.

Pr(no kings) 17296 / 22100 ≈ 0.783

Calculating the Probability of Drawing At Least One King

The probability of drawing at least one king from the 3 cards can be derived by subtracting the probability of drawing no kings from 1.

Pr(at least one king) 1 - Pr(no kings) 1 - 0.783 0.217

Conclusion

Therefore, the probability of drawing at least one king from a standard deck of 52 cards, without replacement, when choosing 3 cards, is approximately 21.7%. This calculation provides an elegant way to understand the underlying principles of probability and combinatorics in card games.

Key Points:- The total number of ways to choose 3 cards from 52 is 22,100.- The total number of ways to choose 3 cards from 48 (no kings) is 17,296.- The probability of drawing no kings is approximately 0.783.- The probability of drawing at least one king is approximately 0.217.

Understanding these calculations can be invaluable for both players and enthusiasts of card games, as well as for students and professionals interested in probability theory.

Related Keywords

Probability of Drawing Kings Standard Deck of Cards Without Replacement