Introduction to the Problem: Drawing Four Cards of the Same Suit

The question of what the probability is when four cards are drawn at random from a standard deck of 52 cards and all four cards belong to the same suit is a classic problem in probability theory. This article aims to provide a detailed, step-by-step explanation of how to solve this problem and to discuss the underlying principles involved. We will also explore the broader context of such problems and their relevance in understanding probability theory.

Understanding the Problem

Let's break down the problem and understand the key elements. A standard deck of 52 cards consists of four suits: Hearts, Diamonds, Clubs, and Spades. Each suit contains 13 distinct cards. The question asks for the probability that, when four cards are drawn at random, all four cards come from the same suit.

Step-by-Step Solution

There are several key steps to solving this problem:

Step 1: Calculate the Total Number of Ways to Choose 4 Cards from 52 Cards

The total number of ways to choose any 4 cards from a deck of 52 cards can be found using the combination formula:

Total ways (binom{52}{4}) (frac{52!}{4!(52-4)!}) (frac{52 times 51 times 50 times 49}{4 times 3 times 2 times 1}) 270725

Step 2: Calculate the Number of Ways to Choose 4 Cards of the Same Suit

Since there are 4 suits, and in each suit there are 13 cards, the number of ways to choose 4 cards from a single suit is:

Ways for one suit (binom{13}{4}) (frac{13!}{4!(13-4)!}) (frac{13 times 12 times 11 times 10}{4 times 3 times 2 times 1}) 715

As there are four suits, the total number of ways to choose 4 cards from one of the suits is:

Total ways 4 (times) 715 2860

Step 3: Calculate the Probability

The probability (P) of drawing 4 cards of the same suit is the ratio of the number of favorable outcomes to the total number of outcomes:

(P) (frac{2860}{270725}) (approx) 0.01057

This means that the probability of drawing four cards of the same suit is approximately 1.057%, or about 0.01057.

Relevance and Applications

Understanding such probability problems is crucial in various fields, including statistics, gaming, cryptography, and data analysis. The steps involved in solving this problem can be adapted to solve similar problems, enhancing one's capability to handle complex probability scenarios.

Conclusion and Further Reading

Probability theory is a fundamental branch of mathematics that finds applications in virtually every aspect of modern life. By understanding how to solve such problems, you can develop a deeper appreciation for the power of probability and its practical implications.

If you are interested in further exploring probability concepts and their applications, consider diving into related topics such as conditional probability, expected value, and statistical inference. These subjects will provide you with a comprehensive understanding of how probability can be both intriguing and practical.