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Chapter 5: Problem 2

Use the multinomial formula and find the probabilities for each. $$ \begin{array}{l}{\text { a. } n=3, X_{1}=1, X_{2}=1, X_{3}=1, p_{1}=0.5, p_{2}=0.3} \\ {p_{3}=0.2} \\ {\text { b. } n=5, X_{1}=1, X_{2}=3, X_{3}=1, p_{1}=0.7, p_{2}=0.2} \\ {p_{3}=0.1} \\ {\text { c. } n=7, X_{1}=2, X_{2}=3, X_{3}=2, p_{1}=0.4, p_{2}=0.5} \\ {p_{3}=0.1}\end{array} $$

Short Answer

Expert verified
a. 0.18, b. 0.0112, c. 0.014

Step by step solution

01

Understanding the Multinomial Formula

The multinomial formula is given by \( P(X_1 = x_1, X_2 = x_2, \ldots, X_k = x_k) = \frac{n!}{x_1!x_2!\ldots x_k!} p_1^{x_1} p_2^{x_2} \ldots p_k^{x_k} \). This formula is used to calculate the probability of a specific distribution of events where each event has a different probability and the total number of events \( n \) is given.
02

Solve for Part a

For part a, apply the multinomial formula with \( n=3, X_1=1, X_2=1, X_3=1, p_1=0.5, p_2=0.3, p_3=0.2 \).Calculate using:\[P = \frac{3!}{1!1!1!} \cdot (0.5)^1 \cdot (0.3)^1 \cdot (0.2)^1\]Calculate factorial and powers:\[3! = 6, \text{thus, } P = 6 \times 0.5 \times 0.3 \times 0.2 = 0.18\].
03

Solve for Part b

For part b, apply the formula with \( n=5, X_1=1, X_2=3, X_3=1, p_1=0.7, p_2=0.2, p_3=0.1 \).Calculate using:\[P = \frac{5!}{1!3!1!} \cdot (0.7)^1 \cdot (0.2)^3 \cdot (0.1)^1\]Calculate factorial and powers:\[5! = 120, 3! = 6, \text{thus, } P = \frac{120}{6} \times 0.7 \times 0.008 \times 0.1 = 0.0112\].
04

Solve for Part c

For part c, apply the formula with \( n=7, X_1=2, X_2=3, X_3=2, p_1=0.4, p_2=0.5, p_3=0.1 \).Calculate using:\[P = \frac{7!}{2!3!2!} \cdot (0.4)^2 \cdot (0.5)^3 \cdot (0.1)^2\]Calculate factorial and powers:\[7! = 5040, 2! = 2, 3! = 6, \text{thus, } P = \frac{5040}{24} \times 0.16 \times 0.125 \times 0.01 = 0.014\].

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Key Concepts

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

Probability Calculations
Probability calculations are essential in understanding the likelihood of different outcomes in an event. In situations where multiple outcomes are possible, like those involving rolling dice or drawing cards, probability calculations help determine the chance of each outcome happening. The multinomial formula is particularly useful for events with multiple possible outcomes, each having its own probability. It helps us predict the frequency of occurrence of different results. For example, calculating the chance of drawing specific colored marbles from a bag. Using the formula involves:
  • Determining the total number of trials or events (e.g., drawing marbles three times).
  • Understanding the probability of each specific outcome (e.g., the probability of drawing a red marble).
  • Applying the formula to get the probability of the chosen distribution of outcomes happening.
Factorials in Statistics
Factorials are a mathematical concept represented by an exclamation point (e.g., 5!). It signifies the product of all positive integers up to a specified number. So, 5! = 5 脳 4 脳 3 脳 2 脳 1. Factorials are vital in statistics, particularly when calculating probabilities for permutations and combinations. Why are they important in the multinomial formula? Factorials help calculate the total possible arrangements of outcomes. For example, with three different events occurring, the factorial tells us the number of ways they can be ordered. In our multinomial formula, we use factorials to estimate the likelihood of outcomes distributing in certain ways, adjusting for how many times each outcome occurs. This is why understanding and being able to calculate factorials is an essential skill in probability and statistics.
Event Distribution
Event distribution refers to how outcomes are spread out in a set of events. Essentially, it's the way outcomes are arranged in a sequence of trials. For example, when tossing a coin twice, the event distribution could be heads-heads, heads-tails, tails-heads, or tails-tails. Understanding how these results distribute allows us to estimate probabilities using methods like the multinomial formula. The multinomial formula accounts for various event distributions by factoring in the number of outcomes and the probability of each outcome. By calculating with the formula, you can predict how frequently you might see a specific distribution of results across multiple events. Event distribution is crucial when dealing with scenarios where several different outcomes can occur simultaneously, making it necessary to predict the likelihood of these distributions in statistics.
Probability Theory
Probability theory is the mathematical framework for quantifying uncertainty. It is the foundation of statistical calculations and helps us understand and predict various phenomena that involve chance. It encompasses rules and concepts, such as probability calculations, distributions, and factorials, allowing us to compute the likelihood of multiple outcomes. This theory is pivotal in fields as diverse as gaming, economics, and even weather forecasting. One key element of probability theory is the law of large numbers, which suggests that as an experiment is repeated numerous times, the experimental probability will approximate the theoretical probability. The multinomial formula, as part of probability theory, helps manage scenarios where multiple outcomes don鈥檛 just have a simple yes/no probability, but vary in likelihood, frequency, and distribution. Understanding and applying probability theory is crucial for evaluating scenarios that involve uncertainty and chance.

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Most popular questions from this chapter

Find the mean, variance, and standard deviation for each of the values of n and p when the conditions for the binomial distribution are met. $$ \begin{array}{l}{\text { a. } n=100, p=0.75} \\ {\text { b. } n=300, p=0.3} \\\ {\text { c. } n=20, p=0.5} \\ {\text { d. } n=10, p=0.8}\end{array} $$

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