Exercises for Chapter 4 (Part A)#
Here are 30 questions designed to assess a student’s ability to differentiate between joint probability, \(P(A \cap B)\), and conditional probability, \(P(A | B)\), based on contextual information.
University Demographics: At a certain university, 30% of students major in business. It’s also noted that 10% of the total student population are female business majors.
Let \(B\) be the event that a student is a business major, and \(F\) be the event that a student is female.
The statement “10% of the total student population are female business majors” translates to which probability expression: \(P(B \cap F)\) or \(P(F | B)\)? Justify your choice.
Answer
This represents \(P(B \cap F) = 0.10\). The phrase “10% of the total student population are female business majors” indicates the intersection of the two events – students who belong to both categories out of the entire population.
Retail Discounts: A retail outlet notes that 45% of its sales are for electronic items. Among all sales, 15% consist of discounted electronics.
Let \(E\) be the event that a sale is for electronics, and \(D\) be the event that a sale is discounted.
The figure “15% of all sales consist of discounted electronics” relates to which probability: \(P(E \cap D)\) or \(P(D|E)\)? Provide the notation and the value.
Answer
This represents \(P(E \cap D)\), specifically \(P(\text{Electronics} \cap \text{Discounted}) = 0.15\). The phrasing “15% of all sales consist of discounted electronics” implies these sales meet both criteria simultaneously out of all possible sales.
Customer Loyalty: A recent survey indicated that 60% of customers were satisfied with a particular service. Among the group of satisfied customers, 90% stated they would recommend the company.
Let \(S\) be the event that a customer was satisfied, and \(R\) be the event that a customer would recommend the company.
The information “Among the group of satisfied customers, 90% stated they would recommend the company” is an example of what type of probability? Express this as \(P(A \cap B)\) or \(P(A | B)\) using the defined events.
Answer
This represents a conditional probability, \(P(R | S) = 0.90\). The phrase “Among the group of satisfied customers” restricts the sample space to only those who were satisfied, which is the hallmark of conditional probability.
Regional Climate: For a specific region, the probability of rain on any day is 0.70. Given that it is raining, the probability of experiencing high humidity is 0.80.
Define events for Rain (R) and High Humidity (H).
Express the information “Given that it is raining, the probability of experiencing high humidity is 0.80” using your defined events and the correct probability notation.
Answer
This represents \(P(H | R) = 0.80\). The phrase “Given that it is raining” explicitly states a condition for the probability of high humidity.
E-Learning Success: A study on online education found that 40% of students choose to enroll in a particular online course. Of the students who enroll, 60% go on to complete the course successfully.
Let \(E\) be the event a student enrolls and \(C\) be the event a student completes the course.
The statement “Of the students who enroll, 60% go on to complete the course successfully” describes what kind of probability? Write it using the events \(E\) and \(C\).
Answer
This represents \(P(C | E) = 0.60\). The phrase “Of the students who enroll” indicates that the 60% is conditioned on enrollment.
Household Assets: In a certain town, 70% of households own a car. It is also known that 35% of all households in this town own both a car and have a garage.
Consider the events \(C\): a household owns a car, and \(G\): a household has a garage.
Translate the statement “35% of all households in this town own both a car and have a garage” into probability notation.
Answer
This represents \(P(C \cap G) = 0.35\). The phrasing “own both a car and have a garage” points to the intersection of the two events relative to all households.
Electoral Demographics: Data from a recent election shows that 55% of the eligible population cast a vote. Furthermore, 30% of the total eligible population both voted and were aged over 65.
Let \(V\) represent the event that a person voted and \(O\) represent the event that a person is over 65.
The information “30% of the total eligible population both voted and were aged over 65” is an example of \(P(V \cap O)\) or \(P(O|V)\)? Specify the correct notation and its value.
Answer
This represents \(P(V \cap O) = 0.30\). The wording “both voted and were aged over 65” out of the “total eligible population” indicates an intersection.
Candidate Screening: A company is reviewing job applications. They find that 80% of applicants possess a college degree. Among those applicants who have a college degree, 70% also have relevant previous work experience.
Let \(D\) be the event an applicant has a college degree, and \(W\) be the event an applicant has previous work experience.
Which probability does the statement “Among those applicants who have a college degree, 70% also have relevant previous work experience” describe: a joint probability or a conditional probability? Provide the specific notation and value.
Answer
This describes a conditional probability, specifically \(P(W | D) = 0.70\). The phrase “Among those applicants who have a college degree” sets a condition for the 70%.
Dining Habits: At a bustling restaurant, 90% of diners select a main course. For diners who order a main course, there’s a 0.50 probability that they will also order an appetizer.
Identify suitable events M (Main Course) and A (Appetizer).
Express the information “For diners who order a main course, there’s a 0.50 probability that they will also order an appetizer” using probability notation.
Answer
This represents \(P(A | M) = 0.50\). The condition “For diners who order a main course” clearly indicates a conditional probability.
Tech Ownership: Surveys show that 60% of individuals own a smartphone. It is also found that 25% of all individuals own both a smartphone and a tablet.
Let \(S\) be the event of owning a smartphone and \(T\) be the event of owning a tablet.
The statistic “25% of all individuals own both a smartphone and a tablet” corresponds to which of the following: \(P(S \cap T)\), \(P(S|T)\), or \(P(T|S)\)? Explain your choice.
Answer
This corresponds to \(P(S \cap T) = 0.25\). The phrase “25% of all individuals own both a smartphone and a tablet” means these individuals possess both devices out of the entire population, indicating an intersection.
Software Adoption: Data indicates that 75% of users have installed the latest software update. Within the group of users who have updated, 85% report satisfaction with the new version.
Let \(U\) be the event a user updated and \(S\) be the event a user reported satisfaction.
The value 85% refers to \(P(S \cap U)\) or \(P(S|U)\)? Write the full expression.
Answer
This represents \(P(S | U) = 0.85\). The phrase “Within the group of users who have updated” establishes a condition.
Horticulture: A plant nursery stocks various plants. Twenty percent of its inventory consists of perennial flowers. Out of all plants in the nursery, 8% are perennial flowers that are also deer-resistant.
Define events \(P\): plant is a perennial flower, and \(D\): plant is deer-resistant.
Which piece of information allows you to write a joint probability? State this probability using your defined events.
Answer
The statement “8% of all its plants are perennial flowers that are also deer-resistant” allows us to write the joint probability \(P(P \cap D) = 0.08\). The key is “also” or “and” applied to the entire stock.
Commuting Choices: A survey on commuting habits finds that 40% of commuters utilize public transport. If a commuter is known to use public transport, there is a 60% likelihood they also own a personal vehicle.
Let \(PT\) be the event a commuter uses public transport and \(PV\) be the event a commuter owns a personal vehicle.
How should the 60% likelihood be expressed in probability notation?
Answer
This should be expressed as \(P(PV | PT) = 0.60\). The condition “If a commuter is known to use public transport” signals a conditional probability.
Library Engagement: Statistics show 50% of library cardholders visit the library at least monthly. Among all cardholders, 20% are monthly visitors who primarily borrow fiction books.
Consider \(M\): cardholder visits monthly, and \(F\): cardholder borrows fiction.
Identify which percentage represents \(P(M \cap F)\) and provide its value.
Answer
The statement “20% of all library cardholders are monthly visitors who primarily borrow fiction books” represents \(P(M \cap F) = 0.20\). The phrasing indicates both conditions are met by this 20% of the total cardholder population.
Household Energy: In a typical household, appliances account for 80% of electricity consumption. Considering only the electricity used by appliances, 30% of that amount is consumed by the refrigerator.
Let \(A\) be the event that electricity is used by an appliance, and \(R\) be the event that electricity is used by the refrigerator.
The 30% figure refers to \(P(R \cap A)\) or \(P(R|A)\)? Explain your reasoning.
Answer
This represents \(P(R | A) = 0.30\). The phrase “Considering only the electricity used by appliances” restricts the context to appliance usage, hence it’s a conditional probability.
Digital Access: It’s reported that 90% of homes have an internet connection. Furthermore, 70% of all homes possess both internet access and a fiber optic connection.
Let \(I\) be having internet access and \(F\) be having a fiber optic connection.
Translate “70% of all homes possess both internet access and a fiber optic connection” into a probability statement.
Answer
This translates to \(P(I \cap F) = 0.70\). The term “both…and” applied to “all homes” indicates an intersection.
Extracurricular Activities: At a high school, 25% of students are members of the debate club. For those students who are in the debate club, 40% also participate in the drama club.
Let \(Debate\) be the event a student is in the debate club, and \(Drama\) be the event a student is in the drama club.
Which probability notation accurately describes the statement “For those students who are in the debate club, 40% also participate in the drama club”?
Answer
This is \(P(\text{Drama} | \text{Debate}) = 0.40\). The condition “For those students who are in the debate club” is key.
Investment Portfolio: An investor’s portfolio consists of 60% stocks. Twenty percent of the investor’s total portfolio value is comprised of stocks from international markets.
Let \(S\) be the event an investment is a stock, and \(I\) be the event an investment is in an international market.
The value “20% of the investor’s total portfolio value is comprised of stocks from international markets” represents which probability: \(P(S \cap I)\) or \(P(I|S)\)?
Answer
This represents \(P(S \cap I) = 0.20\). The phrasing “20% of the investor’s total portfolio… stocks from international markets” (implying stocks AND international) refers to a portion of the entire portfolio satisfying both conditions.
Public Health: In a specific community, 70% of the adult population has received a flu vaccine. Of the vaccinated adults, 95% did not contract the flu during the subsequent season.
Define appropriate events for being vaccinated (V) and contracting the flu (F).
How would you express the information “Of the vaccinated adults, 95% did not contract the flu” using probability notation? (Hint: consider the event “not contracting the flu”).
Answer
Let \(F^c\) be the event of not contracting the flu. The information represents \(P(F^c | V) = 0.95\). The phrase “Of the vaccinated adults” sets the condition.
Product Reliability: A manufacturer observes that 5% of their electronic devices experience a failure within the first year of use. Data also shows that 2% of all devices sold fail in the first year and necessitate a complete replacement.
Let \(F1\) be the event a device fails in the first year, and \(R\) be the event it requires full replacement.
Translate the statement “2% of all devices sold fail in the first year and necessitate a complete replacement” into probability notation.
Answer
This is \(P(F1 \cap R) = 0.02\). The phrasing “…fail in the first year and require a full replacement” applied to “all devices” signifies an intersection.
Home Conveniences: In a survey, 85% of households reported owning a washing machine. Among these households (those with a washing machine), 70% also own a clothes dryer.
Let \(W\) be owning a washing machine and \(D\) be owning a dryer.
The 70% figure is an instance of which type of probability? Write the specific probability statement.
Answer
This is a conditional probability: \(P(D | W) = 0.70\). The context “Among these households (those with a washing machine)” indicates the condition.
Reading Habits: Forty percent of adults subscribe to at least one magazine. Out of all adults, 15% subscribe to a magazine and report reading it thoroughly from cover to cover.
Let \(S\) be subscribing to a magazine, and \(R\) be reading it cover-to-cover.
What does “15% of all adults subscribe to a magazine and report reading it thoroughly” represent in terms of probability notation?
Answer
This represents \(P(S \cap R) = 0.15\). The phrasing “…subscribe…and read…” out of “all adults” indicates an intersection.
Urban Transit: In a metropolitan area, 60% of daily commutes are made using public transportation. If a given commute is by public transportation, there’s a 20% chance that it involves at least one transfer.
Define events \(PT\): commute by public transport, and \(T\): commute involves a transfer.
Express the 20% chance using these events and the correct probability notation.
Answer
This is \(P(T | PT) = 0.20\). The condition “If a given commute is by public transportation” is explicitly stated.
Mobile Technology: Current smartphones are advanced: 90% of them include a camera. Looking at the entire market, 80% of all smartphones sold have both a camera and facial recognition capabilities.
Let \(C\) denote having a camera and \(FR\) denote having facial recognition.
Which piece of information represents \(P(C \cap FR)\)? State its value.
Answer
The information “80% of all smartphones sold have both a camera and facial recognition capabilities” represents \(P(C \cap FR) = 0.80\).
Event Logistics: At a major professional conference, 50% of attendees traveled from out-of-state. For those attendees who came from out-of-state, 70% chose to stay in one of the officially recommended conference hotels.
Let \(OOS\) be the event an attendee is from out-of-state, and \(H\) be the event an attendee stayed in a recommended hotel.
The statement “For those attendees who came from out-of-state, 70% chose to stay…” is an example of what? Provide the probability notation.
Answer
This is an example of conditional probability, \(P(H | OOS) = 0.70\). The phrase “For those attendees who came from out-of-state” clearly defines the condition.
Website Analytics: A popular content website observes that 70% of its daily visitors engage with video content. Across all visitors, 30% both view video content and are subscribed to the site’s newsletter.
Let \(V\) be viewing video content and \(N\) be subscribing to the newsletter.
Determine whether “30% across all visitors both view video content and are subscribed” is \(P(V \cap N)\) or \(P(N|V)\), and provide the value.
Answer
This is \(P(V \cap N) = 0.30\). “Across all visitors” and “both…and” indicate an intersection of events.
Digital Banking: A survey on banking preferences found that 65% of consumers utilize online banking services. Of this group who use online banking, 40% also frequently use a mobile banking application.
Consider \(OB\): uses online banking, and \(MB\): uses a mobile banking app.
Interpret “Of this group who use online banking, 40% also frequently use a mobile banking application” as a probability statement.
Answer
This is \(P(MB | OB) = 0.40\). The condition is “Of this group who use online banking.”
Culinary Preferences: A food survey reveals that 70% of respondents enjoy chocolate. Among all respondents, 25% both enjoy chocolate and specifically prefer dark chocolate.
Let \(C\) be liking chocolate and \(D\) be preferring dark chocolate.
Translate the statement “25% of all respondents both enjoy chocolate and specifically prefer dark chocolate” into the language of probability.
Answer
This translates to \(P(C \cap D) = 0.25\). “Both…and” applied to “all respondents” indicates an intersection.
Academic Progression: Statistics show that 80% of high school graduates pursue some form of higher education. If a graduate decides to pursue higher education, there is a 60% probability they will enroll in a four-year university program.
Let \(HE\) be pursuing higher education and \(U\) be enrolling in a university.
The 60% probability refers to \(P(U \cap HE)\) or \(P(U|HE)\)? Provide the expression.
Answer
This refers to \(P(U | HE) = 0.60\). The condition “If a graduate decides to pursue higher education” is key.
Online User Behavior: On a specific e-commerce website, 95% of users visit the homepage during their session. Overall, 50% of all users to the site visit the homepage and also click on a featured promotional banner.
Define \(H\): user visits homepage, and \(B\): user clicks promotional banner.
The figure “50% of all users to the site visit the homepage and also click on a featured promotional banner” is an example of which probability type (joint or conditional)? Write the notation.
Answer
This is an example of a joint probability, \(P(H \cap B) = 0.50\). The phrasing “visit the homepage and also click” applied to “all users” indicates an intersection.