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Chapter 6: Problem 3

A Gallup survey of 2002 adults found that \(46 \%\) of women and \(37 \%\) of men experience pain daily (San Luis Obispo Tribune, April 6,2000 ). Suppose that this information is representative of U.S. adults. If a U.S. adult is selected at random, are the outcomes selected adult is male and selected adult experiences pain daily independent or dependent? Explain.

Short Answer

Expert verified
The outcomes 'selected adult is male' and 'selected adult experiences pain daily' are dependent because the probability of the adult experiencing pain is influenced by their gender.

Step by step solution

01

Understand the Problem

The problem provides two probabilities: the proportion of women who experience pain and the proportion of men. The question is if the adult selected experiences pain, does it affect the probability of the adult being male or not. That is what it means for two events to be independent.
02

Analyze given Probabilities

We know that \(46 \%\) of women and \(37 \%\) of men experience pain daily. These probabilities are based on gender, meaning that the probability of experiencing pain is different for women than it is for men.
03

Determine if Events are Independent or Dependent

If these were independent events, the probability of experiencing pain would be the same regardless of whether the adult is a man or a woman. However, since the probabilities are different (\(46 \%\) for women and \(37 \%\) for men), it indicates that these events are dependent. That is, the probability of an adult experiencing pain daily may depend on whether the adult is male or female.

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Most popular questions from this chapter

Four students must work together on a group project. They decide that each will take responsibility for a particular part of the project, as follows: $$\begin{array}{lllll} \hline \text { Person } & \text { Maria } & \text { Alex } & \text { Juan } & \text { Jacob } \\ \text { Task } & \begin{array}{l} \text { Survey } \\ \text { design } \end{array} & \begin{array}{l} \text { Data } \\ \text { collection } \end{array} & \text { Analysis } & \begin{array}{l} \text { Report } \\ \text { writing } \end{array} \end{array}$$ Because of the way the tasks have been divided, one student must finish before the next student can begin work. To ensure that the project is completed on time, a timeline is established, with a deadline for each team member. If any one of the team members is late, the timely completion of the project is jeopardized. Assume the following probabilities: 1\. The probability that Maria completes her part on time is 8 . 2\. If Maria completes her part on time, the probability that Alex completes on time is \(.9\), but if Maria is late, the probability that Alex completes on time is only .6. 3\. If Alex completes his part on time, the probability that Juan completes on time is \(.8\), but if Alex is late, the probability that Juan completes on time is only \(.5\). 4\. If Juan completes his part on time, the probability that Jacob completes on time is \(.9\), but if Juan is late, the probability that Jacob completes on time is only .7. Use simulation (with at least 20 trials) to estimate the probability that the project is completed on time. Think carefully about this one. For example, you might use a random digit to represent each part of the project (four in all). For the first digit (Maria's part), \(1-8\) could represent on time and 9 and 0 could represent late. Depending on what happened with Maria (late or on time), you would then look at the digit representing Alex's part. If Maria was on time, \(1-9\) would represent on time for Alex, but if Maria was late, only 1-6 would represent on time. The parts for Juan and Jacob could be handled similarly.

Of the 10,000 students at a certain university, 7000 have Visa cards, 6000 have MasterCards, and 5000 have both. Suppose that a student is randomly selected. a. What is the probability that the selected student has a Visa card? b. What is the probability that the selected student has both cards? c. Suppose that you learn that the selected individual has a Visa card (was one of the 7000 with such a card). Now what is the probability that this student has both cards? d. Are the outcomes has a Visa card and has a MasterCard independent? Explain. e. Answer the question posed in Part (d) if only 4200 of the students have both cards.

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The article "Baseball: Pitching No-Hitters" (Chance [summer 1994]: 24-30) gives information on the number of hits per team per game for all nine-inning major league games played between 1989 and 1993 . Each game resulted in two observations, one for each team. No distinction was made between the home team and the visiting team. The data are summarized in the following table: $$\begin{array}{cc} \begin{array}{c} \text { Hits per team } \\ \text { per game } \end{array} & \begin{array}{c} \text { Number of } \\ \text { observations } \end{array} \\ \hline 0 & 20 \\ 1 & 72 \\ 2 & 209 \\ 3 & 527 \\ 4 & 1048 \\ 5 & 1457 \\ 6 & 1988 \\ 7 & 2256 \\ 8 & 2403 \\ 9 & 2256 \\ 10 & 1967 \\ 11 & 1509 \\ 12 & 1230 \\ 13 & 843 \\ 14 & 569 \\ 15 & 393 \\ >15 & 633 \\ \hline \end{array}$$ a. If one of these games is selected at random, what is the probability that the visiting team got fewer than 3 hits? b. What is the probability that the home team got more than 13 hits? c. Assume that the following outcomes are independent: Outcome 1: Home team got 10 or more hits. Outcome 2: Visiting team got 10 or more hits. What is the probability that both teams got 10 or more hits? d. Calculation of the probability in Part (c) required that we assume independence of Outcomes 1 and 2 . If the outcomes are independent, knowing that one team had 10 or more hits would not change the probability that the other team had 10 or more hits. Do you think that the independence assumption is reasonable? Explain.

Suppose that the following information on births in the United States over a given period of time is available to you: $$\begin{array}{lr} \text { Type of Birth } & \text { Number of Births } \\ \hline \text { Single birth } & 41,500,000 \\ \text { Twins } & 500,000 \\ \text { Triplets } & 5,000 \\ \text { Quadruplets } & 100 \end{array}$$ Use this information to approximate the probability that a randomly selected pregnant woman who reaches full term a. Delivers twins b. Delivers quadruplets c. Gives birth to more than a single child
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