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Magazine Chapter 5: Problem 2
Use the multinomial formula and find the probabilities for each. a. \(n=3, X_{1}=1, X_{2}=1, X_{3}=1, p_{1}=0.5, p_{2}=0.3\) \(\quad p_{3}=0.2\) b. \(n=5, X_{1}=1, X_{2}=3, X_{3}=1, p_{1}=0.7, p_{2}=0.2\) \(\quad p_{3}=0.1\) c. \(n=7, X_{1}=2, X_{2}=3, X_{3}=2, p_{1}=0.4, p_{2}=0.5,\) \(p_{3}=0.1\)
Short Answer
Expert verified
a. 0.18, b. 0.0168, c. 0.021
Step by step solution
01
Understand the Multinomial Formula
The multinomial probability formula is an extension of the binomial distribution. It is used to find the probability for experiments where each trial can result in one of more than two outcomes, given by:\[ P(X_1 = x_1, X_2 = x_2, X_3 = x_3) = \frac{n!}{x_1!x_2!x_3!} p_1^{x_1} p_2^{x_2} p_3^{x_3} \]Here, \(n\) is the total number of trials, \(X_i\) are the number of outcomes in category \(i\), and \(p_i\) are the probabilities of each category.
02
Calculate Probability for Part (a)
Given: \(n=3, X_1=1, X_2=1, X_3=1, p_1=0.5, p_2=0.3, p_3=0.2\).Substitute these into the formula:\[ P(X_1 = 1, X_2 = 1, X_3 = 1) = \frac{3!}{1!1!1!} (0.5)^1 (0.3)^1 (0.2)^1 \]Calculate step-by-step:- \(3! = 6\) and \(1! \times 1! \times 1! = 1\)- Result is: \(6 \times 0.5 \times 0.3 \times 0.2 = 0.18\) Thus, the probability is \(0.18\).
03
Calculate Probability for Part (b)
Given: \(n=5, X_1=1, X_2=3, X_3=1, p_1=0.7, p_2=0.2, p_3=0.1\).Substitute these values into the formula:\[ P(X_1 = 1, X_2 = 3, X_3 = 1) = \frac{5!}{1!3!1!} (0.7)^1 (0.2)^3 (0.1)^1 \]Calculate step-by-step:- \(5! = 120\), and \(1! \times 3! \times 1! = 6\)- Result is: \(\frac{120}{6} \times 0.7 \times 0.008 \times 0.1 = 0.0168\)Thus, the probability is \(0.0168\).
04
Calculate Probability for Part (c)
Given: \(n=7, X_1=2, X_2=3, X_3=2, p_1=0.4, p_2=0.5, p_3=0.1\).Substitute these values into the formula:\[ P(X_1 = 2, X_2 = 3, X_3 = 2) = \frac{7!}{2!3!2!} (0.4)^2 (0.5)^3 (0.1)^2 \]Calculate step-by-step:- \(7! = 5040\), and \(2! \times 3! \times 2! = 24\)- Result is: \(\frac{5040}{24} \times 0.16 \times 0.125 \times 0.01 = 0.021\)Thus, the probability is \(0.021\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Probability Distribution
In statistics, a probability distribution describes how probabilities are distributed over the values of a random variable. When dealing with multinomial distributions, it becomes crucial to understand the nature of probability distributions. These distributions provide a comprehensive way to understand how each outcome of an experiment can occur with certain probabilities.
- Each possible outcome of a statistical experiment is assigned a probability.
- In multinomial probability, we're considering multiple categories, not just binary outcomes.
- The sum of the probabilities of all possible outcomes is always 1.
Multinomial Formula
The multinomial formula is an extension of the binomial formula to experiments with more than two possible outcomes. It provides a structure to calculate probabilities of any scenario with multiple trials and potential results:
\[ P(X_1 = x_1, X_2 = x_2, X_3 = x_3) = \frac{n!}{x_1!x_2!x_3!} p_1^{x_1} p_2^{x_2} p_3^{x_3} \]
This structure helps identify the likelihood of different outcomes happening simultaneously within the given context of fixed probabilities across events.
\[ P(X_1 = x_1, X_2 = x_2, X_3 = x_3) = \frac{n!}{x_1!x_2!x_3!} p_1^{x_1} p_2^{x_2} p_3^{x_3} \]
- n represents the total number of trials.
- X_i denotes the number of occurrences in each category.
- p_i represents the probability of each category.
This structure helps identify the likelihood of different outcomes happening simultaneously within the given context of fixed probabilities across events.
Statistical Experiment
A statistical experiment is a procedure that generates a set of data, primarily designed to understand relationships between variables and predict outcomes. Multinomial experiments are a type of statistical experiment where there are more than two possible outcomes per trial.
- Each experiment trial can result in one of multiple outcomes, such as rolling a die that lands on six possible faces.
- Multinomial experiments are an extension of binomial experiments (which only have two outcomes, like flipping a coin).
- When analyzing statistical experiments, an emphasis is placed on the randomness and large variety of potential outcomes.
Trials and Outcomes
In the context of multinomial probability, trials and outcomes are fundamental components. Each trial represents an instance where an event is observed, producing one of the multiple outcomes defined beforehand.
- In an experiment with multiple trials, the goal is to count how often each outcome occurs.
- Each outcome has a probability attached, indicating the chance it has of occurring during a trial.
- The outcomes are considered mutually exclusive; every trial can result in exactly one of the outcomes.
