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Chapter 5: Problem 86

The article "Obesity, Smoking Damage U.S. Economy," which appeared in the Gallup online Business Journal reported that based on a large representative sample of adult Americans, \(52.7 \%\) claimed that they exercised at least 30 minutes on three or more days per week during \(2015 .\) It also reported that the percentage for millennials (people age \(19-35\) ) was \(57.1 \%,\) and for those over 35 it was \(51.1 \% .\) If an adult American were to be selected at random, are the events selected adult exercises at least 30 minutes three times per week and selected adult is a millennial independent or dependent events? Justify your answer using the given information.

Short Answer

Expert verified
The events "selected adult exercises at least 30 minutes three times per week" and "selected adult is a millennial" are dependent events. This is because the calculated probability of their intersection, \(P(A \cap B)\), is not equal to the product of their individual probabilities, \(P(A) \cdot P(B)\), which is \(P(E) \cdot P(M)\).

Step by step solution

01

Identify the given probabilities and events

We are given the following probabilities: - The probability that an adult American exercises at least 30 minutes on three or more days per week in 2015: \(P(E) = 0.527\) - The probability that a millennial exercises at least 30 minutes on three or more days per week in 2015: \(P(E|M) = 0.571\) - The probability that an adult over 35 exercises at least 30 minutes on three or more days per week in 2015: \(P(E|\overline{M}) = 0.511\) - The events we are interested in determining the independence of are: - A: Selected adult exercises at least 30 minutes three times per week - B: Selected adult is a millennial
02

Calculate the probability of the intersection

To determine if the events A and B are independent, we need to calculate the probability of their intersection, which is denoted as \(P(A \cap B)\). However, we are given conditional probabilities, so we will first calculate the probability of being a millennial (event B) and then use that to find the intersection. In order to find the probability of being a millennial, we can use the law of total probability with the given conditional probabilities: \[P(E) = P(E|M) \cdot P(M) + P(E|\overline{M}) \cdot P(\overline{M})\] Now, we can rearrange this equation to isolate the probability of being a millennial: \[P(M) = \frac{P(E) - P(E|\overline{M}) \cdot P(\overline{M})}{P(E|M) - P(E|\overline{M})}\] And the probability of not being a millennial is: \[P(\overline{M}) = 1 - P(M)\] Finally, we can find the probability of the intersection by plugging in the given values and calculating \(P(A \cap B)\): \[P(A \cap B) = P(E|M) \cdot P(M)\]
03

Check if the events are independent

Now, we can check if the events A and B are independent by comparing the calculated value of \(P(A \cap B)\) to the product of their individual probabilities \(P(A) \cdot P(B)\), which is \(P(E) \cdot P(M)\). If \(P(A \cap B) = P(A) \cdot P(B)\), then the events are independent. If not, the events are dependent. Calculate both products and make the comparison to reach the conclusion.

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Key Concepts

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

Understanding Probability Theory
Probability theory is a branch of mathematics that deals with quantifying the likelihood of events occurring. It provides a framework to predict the outcomes of random processes and helps us to make informed decisions based on uncertain situations.

For instance, when we say that the probability of an adult American exercising at least thrice a week is 52.7%, we're utilizing probability theory to express the chances of this particular event, based on previous data or observations. Within probability theory, we work with numerical values between 0 and 1, with 0 indicating an impossible event and 1 indicating an event that is certain to happen.

As part of probability theory, we can calculate single event probabilities, like the chance of pulling a red card from a deck, to more complex ones like the likelihood of multiple events occurring in tandem, which would require an understanding of concepts such as conditional probability and independence of events.
Breaking Down Conditional Probability
Conditional probability comes into play when the probability of an event is influenced by the occurrence of a previous event. It answers the question, 'How does the probability of an event change if we know that another event has already happened?'

In our exercise, we looked at the probability a millennial exercises, denoted by \(P(E|M)\), indicating the probability of event E (exercising at least 30 minutes three times per week) given that event M (being a millennial) has occurred. This is different from the unconditional probability of exercising, which does not take age into account.

To put it simply, conditional probability helps us refine our predictions by accounting for new information. For example, knowing that someone is a millennial changes the probability that they exercise regularly, based on the data provided.
Deciphering Independence of Events
When discussing independence of events, we are trying to determine whether the occurrence of one event affects the likelihood of another event occurring. Two events are independent if the occurrence of one event does not change the probability of the other event.

In our exercise, we were asked to figure out if being a millennial is independent from the frequency of exercising. To do this, we calculated the probability of both events occurring together, \(P(A \:cap B)\), and compared it to the product of their two individual probabilities \(P(A) \times P(B)\). If these two values are equal, we can conclude that the events are indeed independent.

Understanding event independence is crucial. For example, it can define whether a study's variables are related or not. In product development, it can influence decisions about whether two failures are connected. It is a key concept in probability theory, with profound implications in various real-world contexts.

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Most popular questions from this chapter

The report "Improving Undergraduate Learning" (Social Science Research Council, 2011) summarizes data from a survey of several thousand college students. These students were thought to be representative of the population of all U.S. college students. When asked about a typical semester, \(68 \%\) said they would be taking a class that is reading intensive (requires more than 40 pages of reading per week). Only \(50 \%\) said they would be taking a class that is writing intensive (requires more than 20 pages of writing over the course of the semester). The percentage who said that they would be taking both a reading intensive course and a writing intensive course in a typical semester was \(42 \% .\) In Exercise \(5.40,\) you constructed a hypothetical 1000 table to calculate the following probabilities. Now use the probability formulas of this section to find these probabilities. a. The probability that a randomly selected student would be taking at least one reading intensive or writing intensive course. b. The probability that a randomly selected student would be taking a reading intensive course or a writing intensive course, but not both. c. The probability that a randomly selected student is taking neither a reading intensive nor a writing intensive course.

An appliance manufacturer offers extended warranties on its washers and dryers. Based on past sales, the manufacturer reports that of customers buying both a washer and a dryer, \(52 \%\) purchase the extended warranty for the washer, \(47 \%\) purchase the extended warranty for the dryer, and \(59 \%\) purchase at least one of the two extended warranties. a. Use the given probability information to set up a hypothetical 1000 table. b. Use the table from Part (a) to find the following probabilities: i. the probability that a randomly selected customer who buys a washer and a dryer purchases an extended warranty for both the washer and the dryer. ii. the probability that a randomly selected customer purchases an extended warranty for neither the washer nor the dryer.

5.62 An appliance manufacturer offers extended warranties on its washers and dryers. Based on past sales, the manufacturer reports that of customers buying both a washer and a dryer, \(52 \%\) purchase the extended warranty for the washer, \(47 \%\) purchase the extended warranty for the dryer, and \(59 \%\) purchase at least one of the two extended warranties. In Exercise \(5.34,\) you constructed a hypothetical 1000 table to calculate the following probabilities. Now use the probability formulas of this section to find these probabilities. a. The probability that a randomly selected customer who buys a washer and a dryer purchases an extended warranty for both the washer and the dryer. b. The probability that a randomly selected customer does not purchase an extended warranty for either the washer or dryer.

Eighty-six countries won medals at the 2016 Olympics in Rio de Janeiro. Based on the results: 1 country won more than 100 medals 2 countries won between 51 and 100 medals 3 countries won between 31 and 50 medals 4 countries won between 21 and 30 medals 15 countries won between 11 and 20 medals 15 countries won between 6 and 10 medals 46 countries won between 1 and 5 medals Suppose one of the 86 countries winning medals at the 2016 Olympics is selected at random. a. What is the probability that the selected country won more than 50 medals? b. What is the probability that the selected country did not win more than 100 medals? c. What is the probability that the selected country won 10 or fewer medals? d. What is the probability that the selected country won between 11 and 50 medals?

A man who works in a big city owns two cars, one small and one large. Three- quarters of the time he drives the small car to work, and one-quarter of the time he takes the large car. If he takes the small car, he usually has little trouble parking and so is at work on time with probability 0.9. If he takes the large car, he is on time to work with probability 0.6. Given that he was at work on time on a particular morning, what is the probability that he drove the small car?
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