Probability of Drawing One Club and One King from a Shuffled Deck

In the realm of probability, one intriguing problem revolves around drawing two cards from a standard deck of 52 playing cards and determining the likelihood that one card is a club and the other is a king. This article explores the different methods to calculate this probability, provides a detailed step-by-step solution, and discusses the unique scenarios that arise based on whether the King of Clubs is included or not.

Understanding the Problem

The problem at hand is to calculate the probability of drawing one club and one king from a well-shuffled deck of 52 cards. The order in which the cards are drawn does not matter, and we need to consider all possible outcomes when drawing two cards.

Without the King of Clubs

Let's start by excluding the King of Clubs from our calculations to explore the simpler scenario first.

The total number of ways to draw any two cards from a standard deck is given by the combination formula:

N(s) C(52, 2) 52 * 51 / 2 1326

To find the favorable outcomes, we need to consider two cases:

Case 1: Drawing a club that is not the King of Clubs (12 ways) and drawing a king that is not the King of Clubs (3 ways). Case 2: Drawing the King of Clubs and a club or a king that is not the King of Clubs (15 ways).

Combining these two cases, we get:

P (12 * 3) / 1326 (15) / 1326 36 / 1326 15 / 1326 51 / 1326 1 / 26

With the King of Clubs

Now, let's consider including the King of Clubs in our calculations. This changes the number of favorable outcomes and the total number of combinations.

In this case, we have three configurations:

Case 1: Both cards are not the King of Clubs (3 for kings, 12 for clubs, and 12 * 3 36 ways). Case 2: The first card is the King of Clubs, and the second card is another club or king (1 * 12 1 * 3 15 ways). Case 3: The first card is a club that is not the King of Clubs, and the second card is the King of Clubs (12 * 1 12 ways).

Combining all these cases, we get:

Probability (36 15 12) / 1326 63 / 1326 1 / 21.11

Generalizing the Solution

To provide a more general solution, let's break down the problem step-by-step:

Calculate the total number of ways to draw 2 cards from a deck: C(52, 2) 1326. Identify the favorable outcomes: Case 1: 12 * 3 36 ways (clubs not King of Clubs and kings not King of Clubs). Case 2: 1 * 15 12 * 1 27 ways ( King of Clubs with another club or king, and club with King of Clubs). Combine the cases: P (36 27) / 1326 63 / 1326 1 / 21.11.

Conclusion

In conclusion, the probability of drawing one club and one king from a well-shuffled deck of 52 cards is approximately 0.03846, or about 3.8%. This probability is consistent regardless of whether the King of Clubs is included or not.

Additional Insights

This problem highlights the importance of considering all possible configurations and the impact of unique cards like the King of Clubs. When solving probability problems involving draws from a deck, it's essential to account for all possible scenarios and use appropriate combinatorial methods (combinations and permutations) to ensure accurate results.

Keywords: probability, deck of cards, club