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Chapter 5: Problem 1

Use the multinomial formula and find the probabilities for each. $$ \begin{array}{l}{\text { a. } n=6, X_{1}=3, X_{2}=2, X_{3}=1, p_{1}=0.5, p_{2}=0.3} \\ {p_{3}=0.2} \\ {\text { b. } n=5, X_{1}=1, X_{2}=2, X_{3}=2, p_{1}=0.3, p_{2}=0.6} \\ {p_{3}=0.1} \\ {\text { c. } n=4, X_{1}=1, X_{2}=1, X_{3}=2, p_{1}=0.8, p_{2}=0.1} \\ {p_{3}=0.1}\end{array} $$

Short Answer

Expert verified
Probabilities: a. 0.135, b. 0.0324, c. 0.0096.

Step by step solution

01

Understand the Multinomial Formula

The multinomial formula is used to find probabilities in scenarios where we have multiple categories. For random variables \(X_1, X_2, \ldots, X_k\) with specific probabilities \(p_1, p_2, \ldots, p_k\) respectively, and total trials \(n\), the probability of obtaining \(X_1, X_2, \ldots, X_k\) successes is given by: \[ P(X_1 = x_1, X_2 = x_2, \ldots, X_k = x_k) = \frac{n!}{x_1! x_2! \cdots x_k!} p_1^{x_1} p_2^{x_2} \cdots p_k^{x_k} \]
02

Compute Probability for Case a

For part a, we have \(n=6\), \(X_1=3\), \(X_2=2\), \(X_3=1\), \(p_1=0.5\), \(p_2=0.3\), \(p_3=0.2\). Using the multinomial formula, calculate: \[ P = \frac{6!}{3!2!1!} (0.5)^3 (0.3)^2 (0.2)^1 \] Calculate each part:- \(6! = 720\), \(3! = 6\), \(2! = 2\), \(1! = 1\)- \(\frac{6!}{3!2!1!} = \frac{720}{12} = 60\)- \((0.5)^3 = 0.125\), \((0.3)^2 = 0.09\), \((0.2)^1 = 0.2\)- Multiply terms together: \(60 \times 0.125 \times 0.09 \times 0.2 = 0.135\)
03

Compute Probability for Case b

For part b, we have \(n=5\), \(X_1=1\), \(X_2=2\), \(X_3=2\), \(p_1=0.3\), \(p_2=0.6\), \(p_3=0.1\). Using the formula:\[ P = \frac{5!}{1!2!2!} (0.3)^1 (0.6)^2 (0.1)^2 \]Calculate each part:- \(5! = 120\), \(1! = 1\), \(2! = 2\)- \(\frac{5!}{1!2!2!} = \frac{120}{4} = 30\)- \((0.3)^1 = 0.3\), \((0.6)^2 = 0.36\), \((0.1)^2 = 0.01\)- Multiply terms: \(30 \times 0.3 \times 0.36 \times 0.01 = 0.0324\)
04

Compute Probability for Case c

For part c, we have \(n=4\), \(X_1=1\), \(X_2=1\), \(X_3=2\), \(p_1=0.8\), \(p_2=0.1\), \(p_3=0.1\). Using the formula:\[ P = \frac{4!}{1!1!2!} (0.8)^1 (0.1)^1 (0.1)^2 \]Calculate each part:- \(4! = 24\), \(1! = 1\), \(2! = 2\)- \(\frac{4!}{1!1!2!} = \frac{24}{2} = 12\)- \((0.8)^1 = 0.8\), \((0.1)^1 = 0.1\), \((0.1)^2 = 0.01\)- Multiply terms: \(12 \times 0.8 \times 0.1 \times 0.01 = 0.0096\)

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Key Concepts

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

Probability Calculation
Probability calculation is a fundamental part of learning about the multinomial distribution. This allows us to determine the likelihood of different outcomes in a trial with multiple categories.

When calculating probabilities with the multinomial distribution, you will use a specific formula designed for scenarios where you have more than two possible outcomes per trial.
  • Each outcome has a specific probability. These are noted as \( p_1, p_2, ..., p_k \).

  • The sum of these probabilities must equal 1. This represents a complete set of possible outcomes for a given scenario.

  • We also consider the number of occurrences of each specific outcome, noted by \( X_1, X_2, ..., X_k \).
To find the overall probability, you plug these values into the multinomial formula. The result is a probability value that tells us how likely the specific configuration of outcomes is, given the total number of trials \( n \). Understanding these basics lays the foundation for accurately applying the formula to different scenarios.
Stochastic Processes
Stochastic processes are powerful tools in understanding random phenomena that evolve over time.

They provide insights into how systems change, enabling predictions under uncertainty. In the context of multinomial distribution:
  • Each trial or event in a multinomial can be seen as a step in a stochastic process.

  • The entire process encompasses multiple trials where each trial can lead to one of several different outcomes.

  • Key to these processes is the way probabilities are maintained and utilized over a sequence of trials.
Stochastic processes emphasize the importance of probability transitions between states. This concept helps further divide complex probabilistic scenarios into understandable models, shedding light on how these systems operate dynamically. When you apply multinomial distribution in stochastic frameworks, it allows for tracking and predicting these complex probabilistic transitions.
Discrete Random Variables
Discrete random variables are integral to understanding the multinomial distribution. They allow us to model and predict outcomes where events occur in discrete steps.

Unlike continuous variables, discrete random variables take on distinct, separate values. In this context:
  • The outcomes of a multinomial distribution can be expressed as discrete random variables 鈥� each \( X_i \) represents the count for a particular category.

  • The probabilities associated with these outcomes, \( p_i \), define how likely each count is.

  • The sum of counts \( X_1 + X_2 + \ldots + X_k = n \), where \( n \) is the total number of trials.
Using discrete random variables in the multinomial framework simplifies the complexity, allowing for straightforward probability calculations. This approach is essential for structuring probability models that are based on clear, countable outcomes. By mastering discrete random variables, you open up a deeper understanding of how multinomial distributions can accurately reflect real-world processes in a quantifiable manner.

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Most popular questions from this chapter

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