Calculating the Probability of Selecting 3 Red Balls from an Urn

In probability theory, understanding the likelihood of specific events can be crucial in various scenarios, such as in games, experiments, or statistical analysis. This article delves into the mathematical steps involved in calculating the probability of selecting 3 red balls from a given urn containing different colored balls. We will explore the combination formula and step-by-step reasoning to solve this problem.

Problem Statement

An urn contains 5 red, 3 green, and 4 yellow balls. We are interested in finding the probability of randomly selecting a set of 3 balls such that all of them are red. Let's break down the solution into a series of logical steps.

Step 1: Determine the Total Number of Balls

The urn contains the following balls: 5 red balls 3 green balls 4 yellow balls Therefore, the total number of balls in the urn is:

5 3 4 12

Step 2: Calculate the Total Number of Ways to Choose 3 Balls from 12

The total number of ways to choose 3 balls from 12 can be determined using combinations. The combination formula:

(binom{n}{k} frac{n!}{k!(n-k)!})

For our problem, we need to find the number of ways to choose 3 balls from 12, where n 12 and k 3. Applying the formula:

(binom{12}{3} frac{12!}{3!(12-3)!} frac{12 times 11 times 10}{3 times 2 times 1} 220)

Step 3: Calculate the Number of Ways to Choose 3 Red Balls from the 5 Available

Next, we calculate the number of ways to choose 3 red balls from the 5 available. Using the combination formula again:

(binom{5}{3} frac{5!}{3!(5-3)!} frac{5 times 4}{2 times 1} 10)

Step 4: Calculate the Probability of Choosing 3 Red Balls

The probability P of selecting 3 red balls can be found by dividing the number of ways to choose 3 red balls by the total number of ways to choose 3 balls:

(P(3text{ red balls}) frac{binom{5}{3}}{binom{12}{3}} frac{10}{220} frac{1}{22})

Final Answer

The probability that all balls selected are red is (frac{1}{22}). This result indicates that the event of selecting 3 red balls from the urn is relatively rare, with a probability of only 1 in 22.

Key Takeaways

1. Understanding the total number of balls in the urn is crucial for solving problems involving probability. 2. The combination formula is a powerful tool for calculating the number of ways to choose items from a set without regard to the order of selection. 3. Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Importance in Real-World Applications

Probability calculations like these are often used in scenarios such as quality control in manufacturing, financial risk assessment, and even in everyday games and lottery systems. Understanding these concepts can help in making informed decisions in various fields.