Prison and Plea. Use the data in the

Problem 1CQQ A multiple-choice question on a statistics quiz has possible correct answers of a, b, c, d, e. What is the probability that “false” is the correct answer?

Problem 1CRE: Oscar Winners - Listed below are the differences (in years) between the ages of actresses and actors when they won Oscars. The differences are found by subtracting the age of the male winner from the age of the female winner: (actress age) — (actor age). The differences are found using results from the last 12 years (as listed in Data Set 11 in Appendix B). ?20 ?15 ?3 ?12 6 ?15 ?7 ?9 16 ?18 ?15 ?15 a. Find the mean. Compare the result to the value of the mean that would be expected if there is no gender discrepancy between the ages of Oscar-winning actresses and actors. b. Find the median. Compare the result to the value of the median that would be expected if there is no gender discrepancy between the ages of Oscar-winning actresses and actors. c. Find the standard deviation. d. Find the variance. Be sure to include the units of measurement. e. Find the value of the first quartile, Q1. f. Find the value of the third quartile, Q3. g. Construct a boxplot. What does the boxplot suggest about the distribution of the data?

Problem 2CQQ As the author is creating this exercise, a weather reporter stated that there are 3 chances in 10 of rain today. What is the probability of no rain today?

Prison and Plea. In Exercises 1–10, use the data in the accompanying table (based on data from “Does It Pay to Plead Guilty? Differential Sentencing and the Functioning of the Criminal Courts,” by Brereton and Casper, Law and Society Review, Vol. 16, No. 1). Express all probabilities as decimal numbers. Guilty Plea Plea of Not Guilty Sentenced to Prison 392 58 Not Sentenced to Prison 564 14 Prison and Plea if 1 of the 1028 subjects is randomly selected, find the probability of selecting someone sentenced to prison.

Problem 2CRE Unusual/Unlikely Events a. The mean pulse rate for adult women is 77.5 beats per minute, with a standard deviation of 11.6 beats per minute (based on Data Set 1 in Appendix B). Using the range rule of thumb, would a pulse rate of 100 beats per minute be considered unusual? Explain. b. For the pulse rates of adult women described in part (a), is a pulse rate of 50 beats per minute unusual? Explain. c. For a couple having eight children, is it unlikely to have all girls? Explain. d. For a couple having three children, is it unlikely to have all girls? Explain.

Prison and Plea. In Exercises 1–10, use the data in the accompanying table (based on data from “Does It Pay to Plead Guilty? Differential Sentencing and the Functioning of the Criminal Courts,” by Brereton and Casper, Law and Society Review, Vol. 16, No. 1). Express all probabilities as decimal numbers. Guilty Plea Plea of Not Guilty Sentenced to Prison 392 58 Not Sentenced to Prison 564 14 Prison and Plea Find the probability of being sentenced to prison, given that the subject entered a plea of guilty.

Problem 3CQQ If a day of the week is randomly selected, what is the probability that it is a day containing the letter y?

Problem 3RE: Prison and Plea. Use the data in the accompanying table (based on data from Does It Pay to Plead Guilty? Differential Sentencing and the Functioning of the Criminal Courts, ” by Brereton and Casper, Law and Society Review, Vol. 16, No. 1). Express all probabilities as decimal numbers. Prison and Plea - Find the probability of being sentenced to prison, given that the subject entered a plea of not guilty.

Problem 4CQQ Based on a Harris poll, 20% of adults smoke. If two adults are randomly selected, what is the probability that they both smoke?

Driving Survey In a survey of Americans age 16 years or older, respondents were asked if they texted or e-mailed while driving during the previous 30 days. The results are summarized in the accompanying graph (based on data from AAA Foundation for Traffic Safety). What is wrong with this graph? Do you text or e-mail while diving?

Problem 3CRE Organ Donors USA Today provided information about a survey of 5100 adult Internet users. Of the respondents, 2346 said they are willing to donate organs after death. In this survey conducted for Donate Life America, 100 adults were surveyed in each state and the District of Columbia, and results were weighted to account for the different state population sizes. a. What percentage of respondents said that they are willing to donate organs after death? b. Based on the poll results, what is the probability of randomly selecting an adult who is willing to donate organs after death? c. What term is used to describe the sampling method of randomly selecting 100 adults from each state and the District of Columbia?

Prison and Plea. In Exercises 1–10, use the data in the accompanying table (based on data from “Does It Pay to Plead Guilty? Differential Sentencing and the Functioning of the Criminal Courts,” by Brereton and Casper, Law and Society Review, Vol. 16, No. 1). Express all probabilities as decimal numbers. Guilty Plea Plea of Not Guilty Sentenced to Prison 392 58 Not Sentenced to Prison 564 14 Prison and Plea After comparing the results from Exercises 2 and 3 , what do you conclude about the wisdom of entering a guilty plea?

Problem 5CQQ Estimate the probability that a randomly selected prime-time television show will be interrupted with a news bulletin.

Problem 5CRE Sampling Eye Color Based on a study by Dr. P. Sorita Soni at Indiana University, we know that eye colors in the United States are distributed as follows: 40% brown, 35% blue, 12% green, 7% gray, 6% hazel. a. A statistics instructor collects eye color data from her students. What is the name for this type of sample? b. Identify one factor that might make this particular sample biased and not representative of the general population of people in the United States. c. If one person is randomly selected, what is the probability that this person will have brown or blue eyes? d. If two people are randomly selected, what is the probability that at least one of them has brown eyes?

Prison and Plea. In Exercises 1–10, use the data in the accompanying table (based on data from “Does It Pay to Plead Guilty? Differential Sentencing and the Functioning of the Criminal Courts,” by Brereton and Casper, Law and Society Review, Vol. 16, No. 1). Express all probabilities as decimal numbers. Guilty Plea Plea of Not Guilty Sentenced to Prison 392 58 Not Sentenced to Prison 564 14 Prison and Plea If 1 of the subjects is randomly selected, find the probability of selecting someone who was sentenced to prison or entered a plea of guilty.

Scatterplot of Bear Chest Sizes and Weights. Given below are the chest sizes (inches) and weights (pounds) of 10 bears taken from Data Set 7 in Appendix . Construct a scatterplot. Based on the scatterplot, does there appear to be a correlation between the chest sizes of bears and their weights

In Exercises 6-10, use the following results from the 839 player challenges to referee calls in the first U.S. Open tennis tournament to use the Hawk-Eye electronic instant replay system. Player Challenge Was Accepted Player Challenge Was Rejected Male Player 201 288 Female Player 126 224 If 1 of the 839 challenges is randomly selected, find the probability of getting a challenge that was rejected.

Prison and Plea. In Exercises 1–10, use the data in the accompanying table (based on data from “Does It Pay to Plead Guilty? Differential Sentencing and the Functioning of the Criminal Courts,” by Brereton and Casper, Law and Society Review, Vol. 16, No. 1). Express all probabilities as decimal numbers. Guilty Plea Plea of Not Guilty Sentenced to Prison 392 58 Not Sentenced to Prison 564 14 Prison and Plea If 2 different study subjects are randomly selected, find the probability that they both were sentenced to prison.

In Exercises 6-10, use the following results from the 839 player challenges to referee calls in the first U.S. Open tennis tournament to use the Hawk-Eye electronic instant replay system. Player Challenge Was Accepted Player Challenge Was Rejected Male Player 201 288 Female Player 126 224 If 1 of the 839 challenges is randomly selected, find the probability of getting a challenge that was made by a male player or was rejected.

Prison and Plea. In Exercises 1–10, use the data in the accompanying table (based on data from “Does It Pay to Plead Guilty? Differential Sentencing and the Functioning of the Criminal Courts,” by Brereton and Casper, Law and Society Review, Vol. 16, No. 1). Express all probabilities as decimal numbers. Guilty Plea Plea of Not Guilty Sentenced to Prison 392 58 Not Sentenced to Prison 564 14 Prison and Plea If 2 different study subjects are randomly selected, find the probability that they both entered pleas of not guilty

Problem 7CRE Vermont Hot Lotto Winning the jackpot in the Vermont Hot Lotto game requires that you select five numbers between 1 and 39 and another number between 1 and 19. The first five numbers must match (in any order) the same five numbers that are later drawn, and the sixth number must also match the sixth number that is later drawn. a. What is the probability that the first five selected numbers match the five numbers that are later drawn? b. What is the probability that the sixth selected number matches the sixth number that is later drawn? c. What is the probability of winning the jackpot?

In Exercises 6-10, use the following results from the 839 player challenges to referee calls in the first U.S. Open tennis tournament to use the Hawk-Eye electronic instant replay system. Player Challenge Was Accepted Player Challenge Was Rejected Male Player 201 288 Female Player 126 224 Find the probability of randomly selecting 1 of the 839 challenges and getting a challenge that was accepted and was made by a female player.

Prison and Plea. In Exercises 1-10, use the data in the accompanying table (based on data from "Does it Pay to Plead Guilty? Differential Sentencing and the Functioning of the Criminal Courts," by Brereton and Casper, Law and Society Review, Vol. 16, No. 1). Express all brobabilities as decimal numbers. Guilty Plea Plea of Not Guilty Sentenced to Prison 392 58 Not Sentenced to Prison 564 14 Prison and Plea If 1 of the subjects is randomly selected, find the probability of selecting someone who entered a plea of not guilty or was not sentenced to prison.

In Exercises 6–10, use the following results from the 839 player challenges to referee calls in the first U.S. Open tennis tournament to use the Hawk-Eye electronic instant replay system. Find the probability of randomly selecting 2 different challenges and finding that they were both accepted challenges made by female players.

Prison and Plea. In Exercises 1–10, use the data in the accompanying table (based on data from “Does It Pay to Plead Guilty? Differential Sentencing and the Functioning of the Criminal Courts,” by Brereton and Casper, Law and Society Review, Vol. 16, No. 1). Express all probabilities as decimal numbers. Prison and Plea If 1 of the 1028 subjects is randomly selected, find the probability of selecting someone who was sentenced to prison and entered a guilty plea.

In Exercises 6–10, use the following results from the 839 player challenges to referee calls in the first U.S. Open tennis tournament to use the Hawk-Eye electronic instant replay system. Find the probability of randomly selecting 1 of the 839 challenges and getting a challenge that was accepted, given that it was made by a female player

Prison and Plea. In Exercises 1–10, use the data in the accompanying table (based on data from “Does It Pay to Plead Guilty? Differential Sentencing and the Functioning of the Criminal Courts,” by Brereton and Casper, Law and Society Review, Vol. 16, No. 1). Express all probabilities as decimal numbers. Prison and Plea If 1 of the subjects is randomly selected, find the probability of selecting someone who was not sentenced to prison and did not enter a plea of guilty.

Problem 11E Extending Bayes' Theorem Example in this section included an extension of Bayes’ theorem to include three events, denoted by A, B, C. Write an expression that extends Bayes’ theorem so that it can be used to find P{A |Z), assuming that the initial event can occur in one of four ways: A, B, C, D. Example

Problem 12E Extensions of Bayes' Theorem In Example 1, we used only the initial events of A and A. In Example 2, we used initial events of A, B, and C. If events B and C in Example 2 are combined and denoted as A, we can find P{A \ D) using the simpler format of Bayes’ theorem given in Example 2. How would the resulting value of P{A | D) in Example 2 be affected by using this simplified approach? Example 1 Example 2

Blue Eyes About 35% of the population has blue eyes (based on a study by Dr. P. Sorita Soni at Indiana University). a. If someone is randomly selected, what is the probability that he or she does not have blue eyes? b. If four different people are randomly selected, what is the probability that they all have blue eyes? c. Would it be unlikely to randomly select four people and find that they all have blue eyes? Why or why not?

Composite Sampling for STDs Currently, the rate for sexually transmitted diseases (STDs) is 213 per 100,000 (based on data from the Centers for Disease Control and Prevention). When testing for the presence of STDs, the Acton Medical Testing Company saves money by combining blood samples for tests. The combined sample tests positive if at least one person is infected. If the combined sample tests positive, then the individual blood tests are performed. In a test for STDs, blood samples from 10 randomly selected people are combined. Find the probability that the combined sample tests positive with at least 1 of the 10 people infected. Is it likely that such combined samples test positive?

Georgia Fantasy 5 In the Georgia Fantasy 5 lottery, winning the top prize requires that you select the correct five numbers between 1 and 39 (in any order). What is the probability of winning the top prize? Express the answer as a fraction.

Numbers Game Before the proliferation of legal lotteries, illegal numbers games were commonly run by members of organized crime groups. In the typical numbers game, bettors would win by selecting the same three digits that were later drawn in the same order. A common payoff was $600 for a winning $1 bet. Find the probability of winning with a bet in such a numbers game

Trifecta In horse racing, a trifecta is a bet that the first three finishers in a race are selected in the correct order. In a race with 12 horses, how many different trifecta bets are possible? If you randomly guess the order of the first three finishers in a race with 12 horses, what is the probability of winning?

Problem 1E Pregnancy Test Results. Refer to the results summarized in the table below. a. If one of the 99 test subjects is randomly selected, what is the probability of getting a subject who is pregnant? ________________ b. A test subject is randomly selected and is given a pregnancy test. What is the probability of getting a subject who is pregnant, given that the test result is positive?

Problem 2E Pregnancy Test Results. Refer to the results summarized in the table below. a. One of the 99 test subjects is randomly selected. What is the probability of getting a subject who is not pregnant? ________________ b. A test subject is randomly selected and is given a pregnancy test. What is the probability of getting a subject who is not pregnant, given that the test result is negative?

Problem 3E Survey Results In Orange County, 51% of the adults are males. One adult is randomly selected for a survey involving credit card usage. (See Example) a. Find the prior probability that the selected person is a female. b. It is later learned that the selected survey subject was smoking a cigar. Also, 9.5% of males smoke cigars, whereas 1.7% of females smoke cigars (based on data from the Substance Abuse and Mental Health Services Administration). Use this additional information to find the probability that the selected subject is a female.

Problem 4E Emergency Locator Transmitters An aircraft emergency locator transmitter (ELT) is a device designed to transmit a signal in the case of a crash. The Altigauge Manufacturing Company makes 80% of the ELTs, the Bryant Company makes 15% of them, and the Chartair Company makes the other 5%. The ELTs made by Altigauge have a 4% rate of defects, the Bryant ELTs have a 6% rate of defects, and the Chartair ELTs have a 9% rate of defects. (These are the same results from Example) a. Find the probability of randomly selecting an ELT and getting one manufactured by the Bryant Company. b. If an ELT is randomly selected and tested, find the probability that it was manufactured by the Bryant Company if the test indicates that the ELT is defective.

Problem 5E Emergency Locator Transmitters Use the same ELT data from Exercise. a. Find the probability of randomly selecting an ELT and getting one manufactured by the Chartair Company. b. An ELT is randomly selected and tested. If the test indicates that the ELT is defective, find the probability that it was manufactured by the Chartair Company. Exercise Emergency Locator Transmitters An aircraft emergency locator transmitter (ELT) is a device designed to transmit a signal in the case of a crash. The Altigauge Manufacturing Company makes 80% of the ELTs, the Bryant Company makes 15% of them, and the Chartair Company makes the other 5%. The ELTs made by Altigauge have a 4% rate of defects, the Bryant ELTs have a 6% rate of defects, and the Chartair ELTs have a 9% rate of defects. (These are the same results from Example) a. Find the probability of randomly selecting an ELT and getting one manufactured by the Bryant Company. b. If an ELT is randomly selected and tested, find the probability that it was manufactured by the Bryant Company if the test indicates that the ELT is defective.

Problem 6E Emergency Locator Transmitters Use the same ELT data from Exercise. An ELT is randomly selected and tested. If the test indicates that the ELT is not defective, find the probability that it is from the Altigauge Company. Exercise Emergency Locator Transmitters An aircraft emergency locator transmitter (ELT) is a device designed to transmit a signal in the case of a crash. The Altigauge Manufacturing Company makes 80% of the ELTs, the Bryant Company makes 15% of them, and the Chartair Company makes the other 5%. The ELTs made by Altigauge have a 4% rate of defects, the Bryant ELTs have a 6% rate of defects, and the Chartair ELTs have a 9% rate of defects. (These are the same results from Example) a. Find the probability of randomly selecting an ELT and getting one manufactured by the Bryant Company. b. If an ELT is randomly selected and tested, find the probability that it was manufactured by the Bryant Company if the test indicates that the ELT is defective.

Problem 7E Pleas and Sentences In a study of pleas and prison sentences, it is found that 45% of the subjects studied were sent to prison. Among those sent to prison, 40% chose to plead guilty. Among those not sent to prison, 55% chose to plead guilty. a. If one of the study subjects is randomly selected, find the probability of getting someone who was not sent to prison. b. If a study subject is randomly selected and it is then found that the subject entered a guilty plea, find the probability that this person was not sent to prison.

Problem 8E Pleas and Sentences Use the same data given in Exercise. a. If one of the study subjects is randomly selected, find the probability of getting someone who was sent to prison. b. If a study subject is randomly selected and it is then found that the subject entered a guilty plea, find the probability that this person was sent to prison. Exercise Pleas and Sentences In a study of pleas and prison sentences, it is found that 45% of the subjects studied were sent to prison. Among those sent to prison, 40% chose to plead guilty. Among those not sent to prison, 55% chose to plead guilty. a. If one of the study subjects is randomly selected, find the probability of getting someone who was not sent to prison. b. If a study subject is randomly selected and it is then found that the subject entered a guilty plea, find the probability that this person was not sent to prison.

Problem 9E HIV The New York State Health Department reports a 10% rate of infection with the HIV virus for the “at-risk” population. Under certain conditions, a preliminary screening test for the HIV virus is correct 95% of the time. (Subjects are not told that they are HIV infected until additional tests verify the results.) If someone is randomly selected from the at-risk population, what is the probability that they have the HIV virus if it is known that they have tested positive in the initial screening?

Problem 10E HIV Use the same data from Exercise. If someone is randomly selected from the at-risk population, what is the probability that they have the HIV virus if it is known that they have tested negative in the initial screening? Exercise HIV The New York State Health Department reports a 10% rate of infection with the HIV virus for the “at-risk” population. Under certain conditions, a preliminary screening test for the HIV virus is correct 95% of the time. (Subjects are not told that they are HIV infected until additional tests verify the results.) If someone is randomly selected from the at-risk population, what is the probability that they have the HIV virus if it is known that they have tested positive in the initial screening?

Red Cars Use subjective probability to estimate the probability of randomly selecting a car and getting one that is red.

Problem 13E Biased Coin In an article about confusion of eyewitnesses, John Allen Paulos cites the problem of three coins, one of which is biased so that it turns up heads 75% of the time. If you randomly select one of the coins, toss it three times, and obtain three heads, what is the probability that this is the biased coin?

National Statistics Day a. If a person is randomly selected, find the probability that his or her birthday is October 18, which is National Statistics Day in Japan. Ignore leap years. b. If a person is randomly selected, find the probability that his or her birthday is in October. Ignore leap years. c. Estimate a subjective probability for the event of randomly selecting an adult American and getting someone who knows that October 18 is National Statistics Day in Japan. d. Is it unlikely to randomly select an adult American and get someone who knows that October 18 is National Statistics Day in Japan?