Probability of Drawing Exactly One Queen from a Deck of Cards

The probability of drawing exactly one queen from a standard deck of 52 cards when 2 cards are drawn is a classic problem in probability theory. Understanding this concept is crucial for anyone interested in advanced card games or picking information out of large sets. Let's break down the solution step-by-step.

Step 1: Total Number of Ways to Draw 2 Cards from 52 Cards

The first step is to determine the total number of ways to draw 2 cards from a deck of 52 cards. This is a combination problem where order does not matter. The formula for combinations is:

Combinations formula: $$C(n, k) frac{n!}{k!(n-k)!}$$

Where:

n is the total number of items, which is 52 in this case. k is the number of items to choose, which is 2 in this case.

Solving for our scenario:

Total ways: $$C(52, 2) frac{52!}{2!(52-2)!} frac{52 times 51}{2} 1326$$

Step 2: Number of Favorable Outcomes for Drawing Exactly One Queen

To have exactly one queen in a 2-card hand, we need to choose 1 queen and 1 non-queen card. There are 4 queens and 48 non-queen cards in a standard deck.

The number of ways to choose 1 queen from 4 queens is:

$$C(4, 1) 4$$

The number of ways to choose 1 non-queen card from 48 non-queen cards is:

$$C(48, 1) 48$$

The total number of favorable outcomes is the product of these combinations:

$$C(4, 1) times C(48, 1) 4 times 48 192$$

Step 3: Calculate the Probability

The probability of drawing exactly one queen in a 2-card hand is the ratio of the number of favorable outcomes to the total number of outcomes:

$$P(text{exactly one queen}) frac{192}{1326}$$

For simplicity, this can be simplified:

$$P(text{exactly one queen}) frac{32}{221} approx 0.145$$

Final Answer: $$boxed{frac{32}{221}}$$

Alternative Method for Clarity and Verification

Another approach using combinations involves calculating the number of ways to choose the queen and other cards in all possible positions. For a 2-card draw, the queen can be in any of the 4 positions of the 4-card draw combination, and the other card can be chosen from the remaining 48 non-queen cards.

Combinations formula adjustment:

$$P frac{C(4, 1) times C(48, 1)}{C(52, 2)} frac{4 times 48}{1326} frac{192}{1326} frac{32}{221} approx 0.145$$

Conclusion

Understanding the probability of drawing exactly one queen from a standard deck of 52 cards is essential for various applications, including statistical analysis, probabilistic modeling, and game strategy. By breaking down the problem into smaller, manageable steps and using combinations, we can accurately determine the odds of such an event occurring.