Probability of Drawing One Red and One Blue Ball Without Replacement

Introduction

Understanding the probability of drawing specific combinations of items from a collection can be essential in various fields, including statistics, probability theory, and combinatorics. This article explains how to calculate the probability of drawing one red and one blue ball from a bag containing 5 red balls and 7 blue balls, without replacement.

Understanding the Basics

When drawing from a bag containing 5 red balls and 7 blue balls, the total number of balls is 12. The process of drawing two balls without replacement means the probability of the second draw depends on the result of the first.

Calculating Total Possible Outcomes

The total number of ways to draw 2 balls from 12 can be calculated using the combination formula:

(C(n, k) frac{n!}{k!(n - k)!})

Total ways: (C(12, 2) frac{12!}{2! cdot 10!} frac{12 times 11}{2 times 1} 66)

Calculating Favorable Outcomes

The favorable outcomes, i.e., drawing one red and one blue ball, can be calculated as follows:

Ways to choose 1 red ball from 5: (C(5, 1) 5) Ways to choose 1 blue ball from 7: (C(7, 1) 7)

Total favorable outcomes: (5 times 7 35)

Calculating the Probability

The probability of drawing one red and one blue ball is then calculated as:

(P(1 red , and , 1 blue) frac{35}{66})

Simplifying the Probability

The fraction (frac{35}{66}) cannot be simplified since 35 and 66 have no common factors, thus the final probability is:

(P(1 red , and , 1 blue) frac{35}{66})

Conclusion and Further Applications

This example illustrates the fundamental principles of probability and combinatorics, which are crucial for solving more complex probability problems. The concepts used here can be extended to a variety of real-world scenarios, such as analyzing the chances of winning in games, evaluating risk in financial investments, and understanding biological and ecological processes.

Understanding probability is essential for making informed decisions in data-driven environments, making this topic a cornerstone of modern science, technology, and business analytics.

Related Terms: probability, combinatorics, statistics, combination formula, total outcomes, favorable outcomes, probability theory