Understanding Probability: The Case of Selecting a Red Ball from a Box of 20 White and 15 Red Balls

Introduction to Probability

Probability is the measure of the likelihood that an event will occur and is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. This article will walk you through calculating the probability of selecting a red ball from a box containing 20 white balls and 15 red balls. We will explore both the straightforward method and the more formal combinatorial calculations to arrive at the same result.

The Simple Approach

First, let's consider the straightforward method to calculate the probability of selecting a red ball from this box. This method involves a basic understanding of probability.

There are 20 white balls and 15 red balls, making the total number of balls 35. The probability of selecting a red ball is calculated as the number of red balls divided by the total number of balls. This can be represented as:

Probablity (PRed) Number of Red Balls / Total Number of Balls 15 / 35

This fraction can be simplified:

15 / 35 3 / 7

Alternatively, this probability can be expressed as a decimal: 0.42857142857142855, which is approximately 0.4286.

The Combinatorial Approach

A more formal approach involves the use of combinatorics, which is the branch of mathematics that deals with the study of arrangements and selections. Let's break this down step by step.

Step 1: Counting Total Outcomes

The total number of ways to select any single ball from the 35 balls is represented using combinations (denoted as nCr), where n is the total number of items, and r is the number of items to be chosen. In this case, we want to choose 1 ball out of 35, so the total number of outcomes is:

Total Outcomes 35C1 35

Step 2: Counting Favorable Outcomes

To count the number of favorable outcomes (i.e., selecting a red ball), we use combinations again. We want to choose 1 red ball out of the 15 red balls available, so the number of favorable outcomes is:

Favorable Outcomes 15C1 15

Step 3: Calculating the Probability

Now, the probability of selecting a red ball is the ratio of the number of favorable outcomes to the total number of outcomes:

Required Probability Favorable Outcomes / Total Outcomes 15 / 35

This can be simplified to:

15 / 35 3 / 7

Thus, the probability of selecting a red ball from the box is 3/7 or approximately 0.4286.

Conclusion

Whether you use the simple method or the combinatorial approach, the probability of selecting a red ball from a box containing 20 white and 15 red balls is the same. Both methods provide a clear and systematic way to solve this problem, making probability calculation an essential tool in various real-world applications, from statistical analysis to decision-making in data science.

Keywords: probability calculation, ball selection, combinatorics