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Consider a chance experiment that consists of selecting a customer at random from all people who purchased a car at a large car dealership during 2016 . a. In the context of this chance experiment, give an example of two events that would be mutually exclusive. b. In the context of this chance experiment, give an example of two events that would not be mutually exclusive.

The following table summarizes data on smoking status and age group, and is consistent with summary quantities obtained in a Gallup Poll published in the online article "In U.S., Young Adults' Cigarette Use Is Down Sharply" $$ \begin{array}{|lcc|} \hline & {\text { Smoking Status }} \\ \hline { 2 - 3 } \text { Age Group } & \text { Smoker } & \text { Nonsmoker } \\\ \hline 18 \text { to } 29 & 174 & 618 \\ 30 \text { to } 49 & 333 & 1,115 \\ 50 \text { to } 64 & 384 & 1,445 \\ 65 \text { and older } & 211 & 1,707 \\ \hline \end{array} $$ Assume that it is reasonable to consider these data as representative of the American adult population. Consider the chance experiment or randomly selecting an adult American. a. What is the probability that the selected adult is a smoker? b. What is the probability that the selected adult is under 50 years of age? c. What is the probability that the selected adult is a smoker that is 65 or older? d. What is the probability that the selected adult is a smoker or is age 65 or older?

A study of the impact of seeking a second opinion about a medical condition is described in the paper "Evaluation of Outcomes from a National Patient- Initiated Second-Opinion Program". Based on a review of 6791 patient-initiated second opinions, the paper states the following: "Second opinions often resulted in changes in diagnosis (14.8\%), treatment \((37.4 \%),\) or changes in both \((10.6 \%)\)." Consider the following two events: \(D=\) event that second opinion results in a change in diagnosis \(T=\) event that second opinion results in a change in treatment a. What are the values of \(P(D), P(T),\) and \(P(D \cap T) ?\) b. Use the given probability information to set up a hypothetical 1000 table with columns corresponding to \(D\) and \(D^{C}\) and rows corresponding to \(T\) and \(T^{C}\). c. What is the probability that a second opinion results in neither a change in diagnosis nor a change in treatment? d. What is the probability that a second opinion results is a change in diagnosis or a change in treatment?

a. Suppose events \(E\) and \(F\) are mutually exclusive with \(P(E)=0.64\) and \(P(F)=0.17\) i. What is the value of \(P(E \cap F)\) ? ii. What is the value of \(P(E \cup F)\) ? b. Suppose that \(A\) and \(B\) are events with \(P(A)=0.3, P(B)=0.5\), and \(P(A \cap B)=0.15 .\) Are \(A\) and \(B\) mutually exclusive? How can you tell? c. Suppose that \(A\) and \(B\) are events with \(P(A)=0.65\) and \(P(B)=0.57 .\) Are \(A\) and \(B\) mutually exclusive? How can you tell?

In a small city, approximately \(15 \%\) of those eligible are called for jury duty in any one calendar year. People are selected for jury duty at random from those eligible, and the same individual cannot be called more than once in the same year. What is the probability that an eligible person in this city is selected in both of the next 2 years? All of the next 3 years?

In an article that appears on the website of the American Statistical Association, Carlton Gunn, a public defender in Seattle, Washington, wrote about how he uses statistics in his work as an attorney. He states: I personally have used statistics in trying to challenge the reliability of drug testing results. Suppose the chance of a mistake in the taking and processing of a urine sample for a drug test is just 1 in 100 . And your client has a "dirty" (i.e., positive) test result. Only a 1 in 100 chance that it could be wrong? Not necessarily. If the vast majority of all tests given say 99 in 100 -are truly clean, then you get one false dirty and one true dirty in every 100 tests, so that half of the dirty tests are false. Define the following events as \(T D=\) event that the test result is dirty \(T C=\) event that the test result is clean \(D=\) event that the person tested is actually dirty \(C=\) event that the person tested is actually clean a. Using the information in the quote, what are the values of $$ \text { i. } P(T D \mid D) $$ iii. \(P(C)\) $$ \text { ii. } P(T D \mid C) $$ iv. \(P(D)\) b. Use the probabilities from Part (a) to construct a hypothetical 1000 table. c. What is the value of \(P(T D)\) ? d. Use the information in the table to calculate the probability that a person is clean given that the test result is dirty, \(P(C \mid T D)\). Is this value consistent with the argument given in the quote? Explain.