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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

Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day. What is the probability of getting 35 customers in the first four hours? Assume the store is open 12 hours each day.

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. Six different colored dice are rolled. Of interest is the number of dice that show a one. 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. On average, how many dice would you expect to show a one? e. Find the probability that all six dice show a one. f. Is it more likely that three or that four dice will show a one? Use numbers to justify your answer numerically.

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. The expected number of wins for that upcoming month is: a. 1.67 b. 12 c. \(\frac{382}{1043}\) d. 4.43 Let \(X =\) the number of games won in that upcoming month.

Use the following information to answer the next six exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution. $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline 1 & {0.15} \\ \hline 2 & {0.35} \\ \hline 3 & {0.40} \\ \hline 4 & {0.10} \\ \hline\end{array}$$ What is the probability the baker will sell exactly one batch? \(P(x=1)=\)_________

Identify the mistake in the probability distribution table. $$\begin{array}{|c|c|c|}\hline x & {P(x)} & {x^{*} P(x)} \\ \hline 1 & {0.15} & {0.15} \\ \hline 2 & {0.25} & {0.40} \\ \hline 3 & {0.25} & {0.85} \\\ \hline 4 & {0.20} & {0.85} \\ \hline 5 & {0.15} & {1} \\ \hline\end{array}$$
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