Calculating the Probability of Picking Two Balls Without Replacement: A Comprehensive Guide

Understanding the fundamentals of probability can significantly enhance your analysis of various scenarios. One common type of probability problem involves calculating the probability of picking specific items from a collection without replacement. This article will delve into the calculation of the probability of picking two balls of the same color from an urn containing 5 red, 6 blue, and 4 green balls. We will use combinatorics, a branch of mathematics, to determine this.

Introducing the Problem

The scenario involves an urn with a diverse collection of balls: 5 red, 6 blue, and 4 green. We are interested in calculating the probability of picking two balls of the same color without replacement. This means that once a ball is picked, it is not returned to the urn, thus altering the total number of balls available for the subsequent draw.

Understanding Probability Basics

Probability is a measure of the likelihood of an event occurring. For a single event E, the probability is given by the formula:

where P(E) is the probability of event E, and the number of favorable outcomes is divided by the total number of possible outcomes.

Step-by-Step Calculation

Given the urn contains 5 red, 6 blue, and 4 green balls, we can use the probability formula to calculate the probability of picking two balls of the same color without replacement. Let's break down the process into three parts, each focusing on one color:

Probability of Picking Two Red Balls

Step 1: Calculate the total number of balls in the urn: 5 red 6 blue 4 green 15 balls. Step 2: Determine the probability of picking a red ball on the first draw: 5/15. Step 3: Calculate the probability of picking another red ball on the second draw, given the first was red: (5-1)/(15-1) 4/14. Step 4: Multiply the probabilities from Step 2 and Step 3 to find the probability of both events happening: (5/15) * (4/14) 2/21.

Probability of Picking Two Blue Balls

Step 1: Same total number of balls: 15. Step 2: Probability of picking a blue ball on the first draw: 6/15. Step 3: Probability of picking another blue ball on the second draw, given the first was blue: (6-1)/(15-1) 5/14. Step 4: Multiply the probabilities: (6/15) * (5/14) 1/7.

Probability of Picking Two Green Balls

Step 1: Same total number of balls: 15. Step 2: Probability of picking a green ball on the first draw: 4/15. Step 3: Probability of picking another green ball on the second draw, given the first was green: (4-1)/(15-1) 3/14. Step 4: Multiply the probabilities: (4/15) * (3/14) 2/35.

Conclusion

By carefully applying the principles of probability and combinatorics, we can accurately calculate the likelihood of picking two balls of the same color from the urn without replacement. The calculations for picking two red balls, two blue balls, and two green balls yielded probabilities of 2/21, 1/7, and 2/35, respectively. This approach can be generalized to any similar problem where items are picked without replacement.

Key Takeaways

The probability of an event is the ratio of favorable outcomes to the total number of possible outcomes. Combinatorics helps in calculating probabilities of multiple events occurring in sequence. Without replacement, the total number of possible outcomes changes with each draw.

Further Reading

For more in-depth articles on probability and combinatorics, explore these resources:

Combinatorics: The Basics Probability: Comprehensive Tutorials Statistics and Probability Definitions