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Chapter 4: Problem 15

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time. Construct a PDF table.

Short Answer

Expert verified
Construct a table with event numbers 0-5 and their probabilities: 0.05, 0.05, 0.10, 0.20, 0.25, 0.35.

Step by step solution

01

Understanding the Problem

We need to create a Probability Distribution Function (PDF) table showing the probability of different numbers of events that Javier attends each month.
02

Identifying Possible Outcomes

Javier can attend 0, 1, 2, 3, 4, or 5 events in a month. We list these as the possible outcomes: 0, 1, 2, 3, 4, and 5 events.
03

Assigning Probabilities

We assign probabilities based on the given percentages: 0 events (5%), 1 event (5%), 2 events (10%), 3 events (20%), 4 events (25%), and 5 events (35%).
04

Converting Percentages to Probabilities

Convert the percentage to decimal form to represent the probability: 5% = 0.05, 10% = 0.10, 20% = 0.20, 25% = 0.25, and 35% = 0.35.
05

Constructing the PDF Table

Create the table with columns for 'Number of Events' and 'Probability'. Fill in the table with the corresponding probability for each number of events.

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Key Concepts

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

Probability Table
A probability table is a structured way to display the likelihood of different outcomes in any scenario. In the given exercise, we're looking at how many events Javier volunteers for in a month. A probability table helps us organize this information clearly by showing two main parts:
  • The different numbers of events Javier might attend (from 0 to 5).
  • The probability associated with each of these events.
To set up this table, we define the possible outcomes on one side and the corresponding probabilities on the other. This ensures each outcome has a visual representation of its likelihood. Without such a table, understanding and comparing probabilities can become confusing.
Discrete Outcomes
Discrete outcomes refer to the specific, countable events that can occur in a given situation. In this scenario, Javier can attend 0, 1, 2, 3, 4, or 5 events. These options are discrete because they are distinct and measurable.
  • Discrete refers to countable choices, unlike continuous outcomes which can include any value in an interval.
  • Each outcome is separate; for example, attending 2 events is entirely separate from attending 3 events.
By listing discrete outcomes, we offer a complete view of all the possibilities Javier could face, providing a foundation for assigning probabilities to each.
Percentage Conversion
Converting percentages to probabilities is crucial when dealing with probability distributions since probabilities need to be in decimal form. In the exercise, Javier's volunteering percentages are given: 5%, 10%, 20%, 25%, and 35%.
  • To convert a percentage to a probability, divide by 100. For instance, 5% becomes \(0.05\).
  • This conversion ensures that all probabilities can be added together to make sure they sum to 1, validating the probability distribution.
By converting percentages, we ensure that probabilities are in a standard form required for mathematical operations and comparisons.
Probability Assignment
Probability assignment is the process of associating a numerical probability with each possible outcome. In Javier's case, each number of events he attends in a month is assigned a probability based on the given percentages.
  • This assignment uses the converted probabilities: for example, the probability of attending exactly 5 events becomes \(0.35\).
  • Assigning the right probabilities allows for accurate representation of real-world likelihoods, making it possible to predict how often Javier will attend different numbers of events in a month.
Correctly assigning these probabilities is key to constructing a valid Probability Distribution Function, ensuring each outcome is quantified by how often it can occur.

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Most popular questions from this chapter

According to the South Carolina Department of Mental Health web site, for every 200 U.S. women, the average number who suffer from anorexia is one. Out of a randomly chosen group of 600 U.S. women determine the following. a. In words, define the random variable X. b. List the values that X may take on. c. Give the distribution of \(X, X \sim\) _____(_____,_____) d. How many are expected to suffer from anorexia? e. Find the probability that no one suffers from anorexia. f. Find the probability that more than four suffer from anorexia.

The chance of having an extra fortune in a fortune cookie is about 3%. Given a bag of 144 fortune cookies, we are interested in the number of cookies with an extra fortune. Two distributions may be used to solve this problem, but only use one distribution to solve the problem. a. In words, define the random variable X. b. List the values that X may take on. c. Give the distribution of \(X, X \sim\) _____(_____,_____) d. How many cookies do we expect to have an extra fortune? e. Find the probability that none of the cookies have an extra fortune. f. Find the probability that more than three have an extra fortune. g. As n increases, what happens involving the probabilities using the two distributions? Explain in complete sentences.

Use the following information to answer the next four exercises: Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time. One week is selected at random. Construct a probability distribution table for the data.

Use the following information to answer the next four exercises. Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about 4%. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu. A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 18% of students attend Tet festivities. We are interested in the number of students who will attend the festivities. a. In words, define the random variable \(X.\) b. List the values that \(X\) may take on. c. Give the distribution of \(X . X \sim\) ____(___,____) d. How many of the 12 students do we expect to attend the festivities? e. Find the probability that at most four students will attend. f. Find the probability that more than two students will attend.

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select freshman from the study until you find one who replies 鈥測es.鈥� You are interested in the number of freshmen you must ask. Construct the probability distribution function (PDF). Stop at \(x = 6\). $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline 1 & {} \\ \hline 2 & {} \\\ \hline 3 & {} \\ \hline 4 & {} \\ \hline 5 & {} \\ \hline x & {P(x)} \\\ \hline 6 & {} \\ \hline\end{array}$$
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