Problem 238 - An Introduction to Mathematical Statistics and Its Applications

Question

An urn contains eight red chips, six white chips, and four blue chips. A sample of size 3 is drawn without replacement. Let \(X\) denote the number of red chips in the sample and \(Y\), the number of white chips. Find an expression for \(p_{Y \mid x}(y)\).

Step-by-Step Solution

Answer

The expression for \(p_{Y \mid x}(y)\) can be calculated as \(\frac{\binom{6}{y} \binom{18-x-y}{3-x-y}}{\binom{18-x}{3-x}}\).

Step 1: Recognize the type of problem

This problem is a conditional probability calculation in a multinomial setting. A multinomial setting is where each experiment or trial could have more than two possible outcomes, and each outcome may be categorized into one of multiple categories. One may identify this setting by recognizing that there are more than two types of chips in the urn and that a sample is drawn without replacement, affecting the probabilities of subsequent draws.

Step 2: Understand the variables

Here, \(X\) and \(Y\) are the random variables denoting the number of red chips and white chips drawn, respectively. \(p_{Y \mid x}(y)\) is a conditional probability, asking what the probability of drawing \(y\) white chips is given that \(x\) red chips have already been drawn.

Step 3: Calculate the total number of chips

In total, the urn contains 18 chips (8 red, 6 white, 4 blue). This total figure will be necessary to determine the number of ways to choose remaining chips after certain chips have been drawn.

Step 4: Calculate the conditional probability using combinations

The expression for \(p_{Y \mid x}(y)\) can be calculated as the number of ways to draw \(y\) white chips out of the available pool of chips, divided by the total number of ways to draw the remaining chips after \(x\) red chips have been drawn. This can be represented as \(\frac{\binom{6}{y} \binom{18-x-y}{3-x-y}}{\binom{18-x}{3-x}}\), where \(\binom{n}{k}\) is the binomial coefficient 'n choose k', representing the number of ways to draw \(k\) objects from a pool of \(n\) without order mattering.

Key Concepts

Multinomial Distribution

A multinomial distribution is an extension of the binomial distribution. It applies to scenarios where each trial can result in more than two possible outcomes. Think about drawing different colored chips from an urn, as in our exercise. You're not just checking if you draw a chip or not; instead, you have multiple categories: red, white, and blue chips.
In this type of distribution:

• Each trial is independent.
• There can be more than two outcomes.
• The category probabilities add up to 1.

For our example, with red, white, and blue chips, the outcomes of drawing each color are not independent due to the 'without replacement' condition. Once you draw one chip, it affects the probability of future draws. Hence, multinomial distribution offers a suitable model to understand such processes.

Binomial Coefficient

The binomial coefficient, often read as 'n choose k', is used to determine the number of ways to choose 'k' elements from a set of 'n' elements, without considering the order of the elements. It is written mathematically as \(\binom{n}{k}\) which translates to \[\frac{n!}{k!(n-k)!}.\]
In our exercise, this concept helps us calculate how many different ways we can choose a specific number of white or red chips. For instance, \(\binom{6}{y}\) tells us how many ways we can select 'y' white chips out of 6 available white chips. The binomial coefficient is a fundamental tool in combinatorial calculations because it simplifies the process of counting combinations in probability problems.
The real power of the binomial coefficient comes when applied to problems that involve conditional settings, like in our exercise where determining \(\binom{18-x-y}{3-x-y}\) helps find how the remaining chips can be picked once some have already been chosen.

Combinatorial Analysis

Combinatorial analysis is a strategy in mathematics for counting, arranging, and analyzing arrangements of a set. It provides the mathematical background needed to solve problems involving selection and arrangement possibilities, like those found in our chip-drawing exercise.
When we deal with problems involving different arrangements of items—such as determining how to choose 3 chips from a collection of 18—combinatorial analysis becomes essential. It helps break down the complicated task of counting into more manageable parts using principles such as permutations and combinations.
This kind of analysis allows you to:

• Calculate total possible outcomes.
• Determine probabilities for specific events.
• Solve conditional probability questions efficiently by narrowing down possible scenarios.

For instance, in our exercise, combinatorial analysis is applied in calculating the number of ways to draw red and white chips out of the urn and evaluates how these probabilities change as conditions (like having already picked certain colors) are applied. This overall enables a structured approach to solving probability problems.