Problem 33 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that \(r\) of \(N\) chips are red. Divide the chips into three groups of sizes \(n_{1}, n_{2}\), and \(n_{3}\), where \(n_{1}+\) \(n_{2}+n_{3}=N\). Generalize the hypergeometric distribution to find the probability that the first group contains \(r_{1}\) red chips, the second group \(r_{2}\) red chips, and the third group \(r_{3}\) red chips, where \(r_{1}+r_{2}+r_{3}=r\).

Step-by-Step Solution

Answer

The probability is \(\frac{{{{r}\choose{r_{1}}}{{N-r}\choose{n_{1} - r_{1}}}}}{{{{N}\choose{r}}}}\) for the first group, \(\frac{{{{r-r_{1}}\choose{r_{2}}}{{N-n_{1}-r+r_{1}}\choose{n_{2}-r_{2}}}}}}{{{{N}\choose{r}}}}\) for the second group and \(\frac{{{{r-r_{1}-r_{2}}\choose{r_{3}}}{{N-n_{1}-n_{2}-r+r_{1}+r_{2}}\choose{n_{3}-r_{3}}}}}}{{{{N}\choose{r}}}}\) for the third group. The desired probability is the product of these three probabilities.

Step 1: Understand the Problem

We have \(N\) chips with \(r\) of them being red and we are dividing them into three groups (\(n_{1}, n_{2}\), and \(n_{3}\)) where \(n_{1}+n_{2}+n_{3}=N\). We are asked to find the probability that the first group contains \(r_{1}\) red chips, the second group \(r_{2}\) red chips, and the third group \(r_{3}\) red chips where \(r_{1}+r_{2}+r_{3}=r\).

Step 2: Calculate Combinations for Each Group

We need to calculate how many combinations of red and non-red chips are possible in each group. For group 1, we have \({{r}\choose{r_{1}}}\) ways of selecting \(r_{1}\) red chips from \(r\) red chips and we similarly have \({{N-r}\choose{n_{1} - r_{1}}}\) ways of getting non-red chips. Similarly we can calculate for group 2 and group 3.

Step 3: Calculate Overall Combinations

Now, add up all the ways we can distribute the red chips among the groups and the non-red chips among the groups. We need to calculate \({{N}\choose{r}}\) which is the total number of ways to distribute the \(r\) red chips among the \(N\) chips.

Step 4: Calculate the Probability

Finally, divide the number of desired outcomes (the product of all the combinations for each group calculated in Step 2) by the total number of outcomes (calculated in Step 3) to find the probability.

Key Concepts

Probability Theory

At the heart of the study of randomness and uncertainty is probability theory, a branch of mathematics that quantifies the likelihood of events. Probability calculations are essential when assessing various outcomes in diverse fields like finance, insurance, science, and, of course, in our case, statistics.

In the given exercise, we are considering a scenario where we have a known total number of items (chips) and a subset of these items with a specific feature (red chips), a classic setup for employing probability theory. We want to determine the probability of a particular distribution of this characteristic (redness) among various groups. To calculate this probability, we must consider all the ways the characteristic can be distributed and the conditions that these distributions must satisfy, such as the sum of red chips across all groups equaling the total number of red chips.

Understanding probability theory is crucial for solving problems like this and many others where the outcome is uncertain but can be quantified with mathematical rigor.

Combinatorics

Combinatorics, often considered a part of probability theory, is the mathematics of counting, arranging, and grouping. It's particularly valuable in problems where we need to figure out how many different ways we can organize objects based on certain rules.

To compute the probability in our hypergeometric distribution problem, we harness the power of combinatorics to count all possible combinations. Specifically, we use the combination formula, which in notation looks like \( {{n}\brace{k}} \), representing the number of ways to choose a subset of k elements from a larger set of n elements, without regard to the order.

In each group, we calculate the number of combinations of red and non-red chips. We do this because knowing how many ways we can achieve the desired grouping is necessary before we can assess the probability of that grouping occurring. The principles of combinatorics are thus foundational for solving the problem posed by the exercise.

Statistical Distributions

Statistical distributions are crucial in summarizing the probabilities of different outcomes in a process. In the realm of statistics, various distributions like the normal, binomial, or Poisson distributions help us to model and understand the data and the phenomena they represent.

The problem we are faced with uses a hypergeometric distribution, which is a discrete probability distribution that describes the probability of k successes (in our case, red chips) in n draws, without replacement, from a finite population of size N that contains exactly K successes.

This problem requires us to extend the concept of hypergeometric distribution to three groups. These distributions are vital tools for quantifying the chance occurrences we encounter in statistical analysis. Understanding the properties of hypergeometric distribution enables us to solve the exercise by carefully applying the formulae associated with it, thereby determining the exact probability of our desired outcome.