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Expand Your Knowledge: Multinomial Probability Distribution Consider a multinomial experiment. This means the following: 1\. The trials are independent and repeated under identical conditions. 2\. The outcomes of each trial falls into exactly one of \(k \geq 2\) categories. 3\. The probability that the outcomes of a single trial will fall into ith category is \(p_{i}\) (where \(i=1,2 \ldots, k\) ) and remains the same for each trial. Furthermore, \(p_{1}+p_{2}+\ldots+p_{k}=1\) 4\. Let \(r_{i}\) be a random variable that represents the number of trials in which the outcomes falls into category \(i\). If you have \(n\) trials, then \(r_{1}+r_{2}+\ldots\) \(+r_{k}=n .\) The multinational probability distribution is then $$P\left(r_{1}, r_{2}, \cdots r_{k}\right)=\frac{n !}{r_{1} ! r_{2} ! \cdots r_{2} !} p_{1}^{r_{1}} p_{2}^{(2)} \cdots p_{k}^{r_{2}}$$ How are the multinomial distribution and the binomial distribution related? For the special case \(k=2,\) we use the notation \(r_{1}=r, r_{2}=n-r, p_{1}=p\) and \(p_{2}=q .\) In this special case, the multinomial distribution becomes the binomial distribution. The city of Boulder, Colorado is having an election to determine the establishment of a new municipal electrical power plant. The new plant would emphasize renewable energy (e.g., wind, solar, geothermal). A recent large survey of Boulder voters showed \(50 \%\) favor the new plant, \(30 \%\) oppose it, and \(20 \%\) are undecided. Let \(p_{1}=0.5, p_{2}=0.3,\) and \(p_{3}=0.2 .\) Suppose a random sample of \(n=6\) Boulder voters is taken. What is the probability that (a) \(r_{1}=3\) favor, \(r_{2}=2\) oppose, and \(r_{3}=1\) are undecided regarding the new power plant? (b) \(r_{1}=4\) favor, \(r_{2}=2\) oppose, and \(r_{3}=0\) are undecided regarding the new power plant?

Step by step solution

01

Understanding the Problem

We are given a multinomial distribution scenario with three outcomes for each trial. We want to find the probability of specific outcomes in a sample of 6 voters: (a) 3 favor, 2 oppose, 1 undecided and (b) 4 favor, 2 oppose, 0 undecided.

02

Recall the Multinomial Formula

The multinomial probability distribution formula is given by: \[ P(r_{1}, r_{2}, ext{...}, r_{k}) = \frac{n!}{r_{1}! r_{2}! ext{...} r_{k}!} p_{1}^{r_{1}} p_{2}^{r_{2}} ext{...} p_{k}^{r_{k}} \]where \(n\) is the total number of trials and \(r_{i}\) (where \(i = 1, 2, ..., k\)) represent the number of trials for each category, and \(p_{i}\) is the probability for each category.

03

Calculate Probability for Scenario (a)

Scenario (a) is \(r_{1} = 3\), \(r_{2} = 2\), \(r_{3} = 1\). We apply the formula:\[ P(3, 2, 1) = \frac{6!}{3! 2! 1!} (0.5)^3 (0.3)^2 (0.2)^1 \]Calculate the factorials and powers:- \(6! = 720, 3! = 6, 2! = 2, 1! = 1\),- \((0.5)^3 = 0.125, (0.3)^2 = 0.09, (0.2)^1 = 0.2\).Plug these values into the formula:\[ P(3, 2, 1) = \frac{720}{6 \times 2 \times 1} \times 0.125 \times 0.09 \times 0.2 = 0.135 \].

04

Calculate Probability for Scenario (b)

Scenario (b) is \(r_{1} = 4\), \(r_{2} = 2\), \(r_{3} = 0\). We use the formula:\[ P(4, 2, 0) = \frac{6!}{4! 2! 0!} (0.5)^4 (0.3)^2 (0.2)^0 \]Calculate the factorials and powers:- \(4! = 24, 2! = 2, 0! = 1\),- \((0.5)^4 = 0.0625, (0.3)^2 = 0.09, (0.2)^0 = 1\).Insert these values:\[ P(4, 2, 0) = \frac{720}{24 \times 2 \times 1} \times 0.0625 \times 0.09 \times 1 = 0.03375 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Distribution

The binomial distribution is a special case of the multinomial distribution. When we have just two categories, or possible outcomes, we use the binomial distribution to calculate probabilities. This simplification occurs when the number of categories, denoted as \( k \), equals 2. In a binomial distribution:

• The trials are independent; that is, the outcome of one trial doesn't affect the others.
• Each trial results in one of two possible outcomes, typically referred to as "success" and "failure".
• The probability of success remains constant throughout the trials.

Mathematically, if you have \( n \) independent trials, the binomial probability formula is expressed as:\[P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}\]where \( \binom{n}{r} \) is the binomial coefficient, \( p \) is the probability of success, and \( r \) is the number of successes.
Understanding the binomial distribution can help you grasp more complex probability distributions.

Probability Calculations

Probability calculations are the backbone of statistical methods and are essential to understanding how likely events are to occur. They involve determining the likelihood of a particular outcome happening out of all possible outcomes. In the context of our experiment with Boulder voters, we want to find the probability of certain combinations of opinions - favoring, opposing, or being undecided about the new power plant proposal.We use the formula for the multinomial probability distribution, which expands on the binomial distribution by accounting for more than two categories:\[P(r_1, r_2, \ldots, r_k) = \frac{n!}{r_1! r_2! \ldots r_k!} p_1^{r_1} p_2^{r_2} \ldots p_k^{r_k}\]This formula helps us compute the probability by considering:

• \( n \) as the total number of trials.
• \( r_i \) as the number of attempts falling into category \( i \).
• \( p_i \) as the probability associated with each category.

Practicing these calculations with real-world examples, like election predictions, enhances understanding and application of these concepts.

Statistical Methods

Statistical methods are essential tools for analyzing and interpreting data. They help make informed decisions based on data and probabilities. In our context, understanding statistical methods is crucial to interpreting voter survey results and predicting election outcomes. There are several key statistical methods:

• Descriptive Statistics: Used for summarizing data through mean, median, mode, and standard deviation.
• Inferential Statistics: Allows making predictions or conclusions about a population based on a sample.
• Probability Distributions: Such as the multinomial and binomial distributions, help model the likelihood of different outcomes.

These methods form the foundation of statistical analysis and are widely used in various fields such as economics, business decision-making, and scientific research.
Becoming familiar with these methods enhances your ability to handle data efficiently and draw meaningful insights from complex datasets.