The probability of drawing a King or a Queen from a standard 52-card deck is a common question in probability theory. Let's explore this concept in detail.

Introduction

A standard deck of playing cards contains 52 cards, including 4 Kings and 4 Queens. Understanding the probability of drawing a King or a Queen is essential for players in various card games. Here, we will calculate this probability using the principles of combinatorics and basic probability theory.

Calculating the Probability

The probability of an event is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. In this case, the favorable outcomes are the kings and queens, and the total possible outcomes are all the cards in the deck.

Step 1: Identifying Favorable Outcomes

The number of favorable outcomes is the sum of the number of Kings and the number of Queens, which is 4 4 8.

Step 2: Identifying Total Possible Outcomes

The total possible outcomes are the total number of cards in the deck, which is 52.

Step 3: Calculating the Probability

The probability of drawing a King or a Queen is given by the formula:

Formula:

P(King or Queen) frac{text{Number of Kings and Queens}}{text{Total Number of Cards}} frac{8}{52} frac{2}{13}

Therefore, the probability of drawing a King or a Queen from a well-shuffled deck of 52 cards is (frac{2}{13}) or 15.38%).

Alternative Methods of Calculation

Combinatorial Approach

We can also use combinatorial methods to solve this problem. Let's calculate the number of ways to draw 3 cards that are exclusively Kings and/or Queens.

Step 1: Number of Favorable Outcomes for Drawing 3 Cards

The number of favorable outcomes for drawing 3 cards out of the 8 Kings and Queens is given by the binomial coefficient: (binom{8}{3} frac{8!}{3!(8-3)!} 56)

The number of ways to draw 3 cards from the total deck of 52 cards is also given by the binomial coefficient:(binom{52}{3} frac{52!}{3!(52-3)!} 22100)

Step 2: Probability of Drawing 3 Cards

The probability can now be calculated as:

(text{Answer} frac{binom{8}{3}}{binom{52}{3}} frac{56}{22100} frac{14}{5525})

This shows that the probability remains the same when considering a smaller set of favorable outcomes.

Sequential Probability Calculation

Another way to calculate the probability is by considering the sequential drawing of cards.

First Card

The probability of the first card being a King or a Queen is:

(frac{8}{52} frac{2}{13})

Second Card

If the first card drawn is a King or a Queen, the probability of the second card being a King or a Queen is:

(frac{8}{52} times frac{7}{51} frac{56}{2652} frac{14}{663} approx 0.02123)

Third Card

If the first two cards drawn are Kings or Queens, the probability of the third card being a King or a Queen is:

(frac{8}{52} times frac{7}{51} times frac{6}{50} approx 0.0025333966515 approx 0.25%)

These calculations confirm the initial probability as well.

Conclusion

The probability of drawing a King or a Queen from a standard 52-card deck remains consistent at (frac{2}{13} approx 15.38%). Whether using combinatorial methods, sequential probability, or basic probability theory, the result is the same.