When playing with a standard deck of 52 cards, the question of how often a pair will be drawn from five cards arises frequently, especially in games like poker. This article aims to explore and calculate the probability of drawing a pair from five cards. We will first determine the probability of not drawing a pair and then use it to find the probability of at least one pair. Our analysis includes detailed calculations, combinatorial methods, and examples for clarity.

Introduction

Welcome to the world of combinatorial analysis and probability theory. In this article, we will delve into the intricacies of calculating the probability of drawing a pair when five cards are dealt from a standard deck of 52 cards. Whether you're a seasoned poker player or a beginner, understanding this concept can add a new layer of enjoyment and strategic depth to your games.

Calculating the Probability of Not Drawing a Pair

To start, we will determine the probability of not drawing a pair when five cards are dealt from a deck. The key to solving this is to calculate the probability of drawing five cards that do not form a pair. Let's break down the steps:

For the first card, any of the 52 cards can be drawn. There are no restrictions yet, so the probability is 1 (100%).

The probability that the second card does not pair with the first card is 48/51. This is because out of the remaining 51 cards, 48 do not share the same rank as the first card.

The probability that the third card does not pair with the first two cards is 44/50. Here, out of the remaining 50 cards, 44 do not share the same rank as either of the previous two cards.

The probability that the fourth card does not pair with the first three cards is 40/49. Out of the remaining 49 cards, 40 do not share the same rank as any of the previous three cards.

The probability that the fifth card does not pair with the first four cards is 36/48. Out of the remaining 48 cards, 36 do not share the same rank as any of the previous four cards.

The overall probability of not drawing a pair is the product of these individual probabilities:

[text{Probability (no pairs)} frac{48}{51} times frac{44}{50} times frac{40}{49} times frac{36}{48}]

(text{Probability (no pairs)} approx 0.60874612048)

Calculating the Probability of At Least One Pair

The probability of drawing at least one pair is the complement of the probability of not drawing a pair. Therefore, we can calculate it as follows:

[text{Probability (at least one pair)} 1 - text{Probability (no pairs)}][text{Probability (at least one pair)} 1 - 0.60874612048 approx 0.39125387952]

So, the probability of drawing at least one pair when five cards are dealt from a deck is approximately 0.391 or 39.1%.

Alternative Combinatorial Approach

Besides the step-by-step approach, we can use combinatorial methods to find the probability. Let's consider the following steps:

Total number of ways to choose 5 cards from 52: (52 choose 5)

Number of ways to choose 2 pairs and 1 non-paired card:

Number of ways to choose 2 ranks out of 13 for the pairs: (13 choose 2)

Number of ways to choose 2 suits for each pair: (4 choose 2) for each pair

Number of ways to choose the rank for the non-paired card: 11 (since 13 - 2 pairs 11 ranks left)

Number of ways to choose 1 suit for the non-paired card: 4

The total number of favorable outcomes is: (13 choose 2) (times 4 choose 2 times 4 choose 2) (times 11 times 4) (times frac{5!}{2!2!1!} 123,552)

The total number of possible 5-card hands: (52 choose 5)

The probability of drawing a pair using this method is:

[text{Probability (at least one pair)} frac{123,552}{2598960} approx 0.4754]

This approach yields a slightly higher probability, approximately 0.4754 or 47.54%, which can be attributed to the inclusion of more detailed combinatorial configurations.

Conclusion

In conclusion, the probability of drawing at least one pair when five cards are drawn from a deck of 52 cards is approximately 39.1% using the step-by-step approach and about 47.54% using the combinatorial method. Both methods provide valuable insights into the probabilities involved in card games and highlight the importance of thorough mathematical analysis in understanding such probabilities.