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Chapter 5: Problem 25

A survey asks subjects whether they believe that global warming is happening (yes or no) and how much fuel they plan to use annually for automobile driving in the future, compared to their past use (less, about the same, more). a. Show the sample space of possible outcomes by drawing a tree diagram that first gives the response on global warming and then the response on fuel use. b. Let \(A\) be the event of a "yes" response on global warming and let \(\mathrm{B}\) be the event of a "less" response on future fuel use. Suppose \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})>\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B}) .\) Indicate whether \(\mathrm{A}\) and \(\mathrm{B}\) are independent events, and explain what this means in nontechnical terms.

Short Answer

Expert verified
A and B are not independent. Believing in global warming affects plans for reducing fuel usage.

Step by step solution

01

Identify Survey Responses for Global Warming

The survey asks subjects' opinions on global warming, which can be a 'yes' or 'no' response. These are the two possible outcomes for global warming.
02

Identify Survey Responses for Future Fuel Use

The survey asks about the future fuel use compared to past usage, with options being 'less', 'about the same', or 'more'. These are the three potential responses for fuel use.
03

Develop the Tree Diagram

Start by listing the initial branches for global warming: 'yes' or 'no'. From each of these branches, add further branches for the responses to fuel use: 'less', 'about the same', 'more'. This leads to a total of six possible outcomes. The tree diagram would look like: - Yes: - Less - About the same - More - No: - Less - About the same - More
04

Define Events A and B

Event \( A \) corresponds to a 'yes' response for global warming. Event \( B \) means the subject plans to use less fuel in the future.
05

Analyze Independence of Events

Two events are independent if the probability of both events occurring together, \( P(A \text{ and } B) \), is equal to the product of their individual probabilities, \( P(A) \times P(B) \). Here, \( P(A \text{ and } B) > P(A) \times P(B) \) implies that knowing the outcome of \( A \) affects the probability of \( B \), meaning \( A \) and \( B \) are not independent.
06

Explain Implications in Nontechnical Terms

In layman's terms, the outcome of a 'yes' response to global warming affects the likelihood that a person will say they plan to use less fuel in the future. There's a relationship between believing in global warming and intending to reduce fuel usage.

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Key Concepts

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

Sample Space
In probability theory, the concept of sample space is fundamental. A sample space includes all the possible outcomes of an experiment. In our survey example, respondents can answer in a variety of ways regarding global warming and future fuel use.
For global warming, they can either respond 'yes' or 'no', while for fuel use, they can choose from 'less', 'about the same', or 'more'.
Thus, by coupling each response regarding global warming with each of the responses about fuel use, we uncover every possible outcome. This complete set of outcomes is what we call the sample space.
  • The sample space for global warming consists of 'yes' or 'no'.
  • The sample space for fuel usage includes 'less', 'about the same', and 'more'.
By evaluating each potential pairing, you'll see there are six possible combinations: 'yes-less', 'yes-about the same', 'yes-more', 'no-less', 'no-about the same', and 'no-more'. Each combination represents a distinct path in the sample space.
Tree Diagram
A tree diagram is a visual tool for organizing and depicting all potential outcomes of an event. It starts with a single point from which various branches extend, each representing a possible decision or outcome. This diagram grows and branches out until all possibilities are covered.
In our exercise, the tree diagram begins with branches for the responses to global warming: 'yes' or 'no'. From each of these initial branches, we draw further branches to account for the options regarding fuel use: 'less', 'about the same', or 'more'. Thus, every path from start to finish on the tree corresponds to a unique combination of the two sets of responses.
  • Initial branches: responses to global warming ('yes', 'no').
  • Secondary branches: responses to fuel use ('less', 'about the same', 'more').
By systematically branching from one decision point to the next, tree diagrams help you visually track all possible outcomes in clear, sequential order. This method is particularly helpful when calculating probabilities for more complex scenarios.
Independent Events
In probability, events are considered independent if the occurrence of one event does not affect the probability of the other occurring. Mathematically, events A and B are independent if:\[ P(A \text{ and } B) = P(A) \times P(B) \]In our scenario, event A is a 'yes' response on global warming, and event B is planning to use 'less' fuel in the future.
When we find that \( P(A \text{ and } B) > P(A) \times P(B) \), it implies that the likelihood of both events happening together is greater than if they were independent. This indicates that a belief in global warming significantly influences one's decision to use less fuel.
In practical terms, this means that believing in global warming might encourage a person to decrease their fuel consumption. These events are not standalone; they are interconnected, illustrating how beliefs can influence actions.

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Most popular questions from this chapter

Every year the insurance industry spends considerable resources assessing risk probabilities. To accumulate a risk of about one in a million of death, you can drive 100 miles, take a cross country plane flight, work as a police officer for 10 hours, work in a coal mine for 12 hours, smoke two cigarettes, be a nonsmoker but live with a smoker for two weeks, or drink 70 pints of beer in a year (Wilson and Crouch, \(2001,\) pp. \(208-209\) ). Show that a risk of about one in a million of death is also approximately the probability of flipping 20 heads in a row with a balanced coin.

Larry Bird, who played pro basketball for the Boston Celtics, was known for being a good shooter. In games during \(1980-1982,\) when he missed his first free throw, 48 out of 53 times he made the second one, and when he made his first free throw, 251 out of 285 times he made the second one. a. Form a contingency table that cross tabulates the outcome of the first free throw (made or missed) in the rows and the outcome of the second free throw (made or missed) in the columns. b. For a given pair of free throws, estimate the probability that Bird (i) made the first free throw and (ii) made the second free throw. c. Estimate the probability that Bird made the second free throw, given that he made the first one. Does it seem as if his success on the second shot depends strongly, or hardly at all, on whether he made the first?

Pro basketball player Shaquille O'Neal is a poor free-throw shooter. Consider situations in which he shoots a pair of free throws. The probability that he makes the first free throw is \(0.50 .\) Given that he makes the first, suppose the probability that he makes the second is \(0.60 .\) Given that he misses the first, suppose the probability that he makes the second one is 0.40 . a. What is the probability that he makes both free throws? b. Find the probability that he makes one of the two free throws (i) using the multiplicative rule with the two possible ways he can do this and (ii) by defining this as the complement of making neither or both of the free throws. c. Are the results of the free throws independent? Explain.

Part of a student opinion poll at a university asks students what they think of the quality of the existing student union building on the campus. The possible responses were great, good, fair, and poor. Another part of the poll asked students how they feel about a proposed fee increase to help fund the cost of building a new student union. The possible responses to this question were in favor, opposed, and no opinion. a. List all potential outcomes in the sample space for someone who is responding to both questions. b. Show how a tree diagram can be used to display the outcomes listed in part a.

For a family with two children, let A denote \\{first child is female\\}, let B denote (at least one child is female\\}, and let \(C\) denote \\{both children are female\\}. a. Show that \(\mathrm{P}(\mathrm{C} \mid \mathrm{A})=1 / 2\). b. Are \(A\) and \(C\) independent events? Why or why not? c. Find \(\mathrm{P}(\mathrm{C} \mid \mathrm{B})\). d. Describe what makes \(\mathrm{P}(\mathrm{C} \mid \mathrm{A})\) different than \(\mathrm{P}(\mathrm{C} \mid \mathrm{B})\).
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