Calculating the Probability of Drawing Two Balls of the Same Color from an Urn

This article delves into the problem of calculating the probability of drawing two balls of the same color from an urn. Combinatorics offers a precise and efficient way to approach this kind of probabilistic problem. We will explore how to calculate the probability using combinations and provide a step-by-step solution.

The Problem Statement

Imagine an urn that contains 4 blue and 6 red balls. The goal is to find the probability of drawing two balls such that they are of the same color. This problem involves basic probability theory and combinatorics, and it is a common exercise in introductory statistics courses. Let's break down the solution step by step.

Understanding the Problem

We are dealing with an urn containing 4 blue balls and 6 red balls, making a total of 10 balls. We need to draw 2 balls from this urn. The problem can be divided into two parts: drawing two blue balls and drawing two red balls. We'll sum the probabilities of these two events to find the overall probability.

Calculating the Probability for Each Case

To calculate the probability of drawing two balls of the same color, we need to use the concept of combinations. Combinations are used when the order of selection does not matter. The formula for combinations is given by:

C(n, k) n! / (k!(n-k)!)

where n is the total number of items, and k is the number of items to be chosen.

Probability of Drawing Two Blue Balls

The number of ways to choose 2 blue balls from 4 blue balls is given by:

C(4, 2) 4! / (2!(4-2)!) 6

The number of ways to choose 2 balls from 10 balls is given by:

C(10, 2) 10! / (2!(10-2)!) 45

Therefore, the probability of drawing two blue balls is:

P(2 blue balls) C(4, 2) / C(10, 2) 6 / 45 0.1333

Probability of Drawing Two Red Balls

The number of ways to choose 2 red balls from 6 red balls is given by:

C(6, 2) 6! / (2!(6-2)!) 15

Using the same calculation for the total number of ways to choose 2 balls from 10 balls:

C(10, 2) 45

Therefore, the probability of drawing two red balls is:

P(2 red balls) C(6, 2) / C(10, 2) 15 / 45 0.3333

Combining the Probabilities

Since the two events (drawing two blue balls or drawing two red balls) are mutually exclusive, the total probability of drawing two balls of the same color is the sum of the individual probabilities:

P(same color) P(2 blue balls) P(2 red balls) 0.1333 0.3333 0.4666

Conclusion

By utilizing combinatorial methods and basic probability principles, we have calculated that the probability of drawing two balls of the same color from an urn containing 4 blue and 6 red balls is 0.4666. This solution not only provides a clear understanding of the problem but also offers a foundational example of how to apply combinatorics in real-world probability problems.

Related Topics and Resources

For further exploration, readers may want to delve into the following related topics:

Combinatorics: Understanding the principles of combinations and permutations is crucial for solving a wide variety of probability problems. Probability in Statistics: This problem is a fundamental example in statistics, particularly in the study of random events and statistical inference. Examples of Probability in Real Life: Exploring real-life scenarios where similar probability calculations are applied can enhance understanding.

For those interested in learning more, resources such as online courses, textbooks, and tutorials on probability and combinatorics can be excellent starting points.