Probability Distribution in a Random Ball Drawing Experiment

Imagine a scenario in which a bag contains 5 white balls, 4 red balls, 3 blue balls, and 2 yellow balls. We are to draw a ball at random, replace it, and then repeat the process for a total of 10 draws. The objective is to calculate the probability of obtaining exactly 3 white, 3 red, 2 blue, and 2 yellow balls.

Understanding the Problem

This problem can be comprehensively solved by applying the principles of multinomial distribution, a fascinating topic that generalizes the binomial distribution to the case of multiple outcomes. In this specific scenario, each draw can result in one of four outcomes (a white, red, blue, or yellow ball), with the process being repeated 10 times.

Calculating the Probability

The formula for multinomial distribution is given by:

[ P(X_1 x_1, X_2 x_2, ldots, X_k x_k) frac{n!}{x_1!x_2!cdots x_k!}p_1^{x_1}p_2^{x_2}cdots p_k^{x_k} ]

where:

(n) is the total number of trials (in this case, 10 draws), (x_1, x_2, ldots, x_k) are the observed counts for each of the (k) categories (in this case, white, red, blue, and yellow balls), and (p_1, p_2, ldots, p_k) are the probabilities of each category, given that they sum to 1.

Application of the Multinomial Distribution

Given the categories and their corresponding probabilities, we can substitute the given values into the multinomial formula:

[ P(3 text{ white}, 3 text{ red}, 2 text{ blue}, 2 text{ yellow}) frac{10!}{3!3!2!2!} left(frac{5}{14}right)^3 left(frac{4}{14}right)^3 left(frac{3}{14}right)^2 left(frac{2}{14}right)^2 ]

Breaking this down further:

(n 10) (x_1 3) (white balls), (x_2 3) (red balls), (x_3 2) (blue balls), (x_4 2) (yellow balls), (p_1 frac{5}{14}) (probability of a white ball), (p_2 frac{4}{14}) (probability of a red ball), (p_3 frac{3}{14}) (probability of a blue ball), (p_4 frac{2}{14}) (probability of a yellow ball).

Each of these probabilities are derived from the distribution of the balls in the bag. Since the draws are with replacement, the probabilities remain constant throughout the process.

Theoretical and Practical Implications

This problem is not only a practical exercise in probability but also a great example of how theoretical concepts can be applied to real-world scenarios. In a broader context, the use of multinomial distribution can be extended to more complex and varied problems in statistics and probability, such as analyzing data from multiple sources or modeling situations with multiple outcomes.

Conclusion

Understanding and applying the multinomial distribution to real-world problems provides valuable insights into the underlying principles of probability. By breaking down the problem into manageable steps and using the appropriate formula, we can accurately calculate probabilities that are essential for making informed decisions.

Remember, practice forms the foundation of mastering any concept. Whether it's 45 years ago or today, the key to success lies in diligent practice and a willingness to apply what you learn in new contexts.

Keywords

- multinomial distribution - random ball drawing - probability calculation