Probability of Drawing 2 Red Balls from a Mixture of Colored Balls

When dealing with combinatorial problems, such as calculating the probability of drawing specific balls from a mixed collection, it's crucial to employ appropriate mathematical methods. In this article, we will explore the probability of drawing 2 red balls from a total collection of 6 white, 5 red, and 3 orange balls. This will be achieved through a step-by-step application of combinatorial methods and binomial coefficients.

Overview of the Problem

We are interested in determining the probability of drawing exactly 2 red balls when we randomly select 4 balls from a collection of 6 white, 5 red, and 3 orange balls. Let's break down the steps to find this probability.

Total Number of Balls and the Total Number of Ways to Choose 4 Balls

The total number of balls in the collection is 14 (6 white 5 red 3 orange). We need to calculate the total number of ways to choose 4 balls from these 14 balls. This can be represented using the binomial coefficient:

total ways (binom{14}{4} frac{14 times 13 times 12 times 11}{4 times 3 times 2 times 1} 1001)

Desired Outcome: 2 Red Balls and 2 Non-Red Balls

Our goal is to determine the number of favorable outcomes where we draw exactly 2 red balls and the remaining 2 balls are non-red (i.e., either white or orange). There are 5 red balls, and we need to select 2 red balls out of these 5. The number of ways to do this is:

ways to choose 2 red (binom{5}{2} frac{5 times 4}{2 times 1} 10)

Next, we need to choose 2 non-red balls from the remaining 9 balls (6 white 3 orange). The number of ways to do this is:

ways to choose 2 non-red (binom{9}{2} frac{9 times 8}{2 times 1} 36)

The total number of favorable outcomes is the product of these two results:

favorable outcomes (10 times 36 360)

Calculating the Probability

To find the probability, we use the ratio of the number of favorable outcomes to the total number of possible outcomes:

P(2 red) frac{360}{1001} ≈ 0.359

This can be expressed as:

P(2 red) ≈ 0.36 or 35.9%

Conclusion

The probability of drawing exactly 2 red balls when drawing 4 balls from a collection of 6 white, 5 red, and 3 orange balls is approximately 0.359. This calculation involves a detailed understanding of combinatorial methods and the use of binomial coefficients to determine the desired probabilities.

Further Reading

For a deeper understanding of combinatorial methods and binomial coefficients, you may explore resources on discrete mathematics, probability theory, and statistics. Understanding these concepts will help in solving a wide range of probability problems in various real-world scenarios.