Expected Value for a Magazine Sweepstakes Reader’s Digest

Problem 1BSC Notation In analyzing hits by V-l buzz bombs in World War II, South London was partitioned into 576 regions, each with an area of 0.25 km2. A total of 535 bombs hit the combined area of 576 regions. Assume that we want to find the probability that a randomly selected region had exactly two hits. In applying Formula 5-9, identify the values of µ, x, and e. Also, briefly describe what each of those symbols represents.

Problem 1CQQ Is a probability distribution defined if the only possible values of a random variable are 0, 1, 2, 3, and P(0) = P(l) = P(2) = P(3) = 0.25?

Problem 1CRE Weekly Instruction Time The Organization for Economic Cooperation and Development provided the following mean weekly instruction times (hours) for elementary and high school students in various countries: 22.2 (United States); 24.8 (France); 24.2 (Mexico); 26.9 (China); 23.8 (Japan). Use the five given times for the following. a. Find the mean. ________________ b. Find the median. ________________ c. Find the range. ________________ d. Find the standard deviation. ________________ e. Find the variance. ________________ f. Use the range rule of thumb to identify the range of usual values. ________________ g. Based on the result from part (f), are any of the times unusual? Why or why not? ________________ h. What is the level of measurement of the data: nominal, ordinal, interval, or ratio? ________________ i. Are the data discrete or continuous? ________________ j. There is something fundamentally wrong with using the given times to find statistics such as the mean. What is wrong?

Problem 1RE In Exercises, assume that 40% of the population has brown eyes (based on data from Dr. P. Sorita at Indiana University). Brown Eyes If six people are randomly selected, find the probability that none of them has brown eyes.

Problem 2BSC Tornadoes During a recent 46-year period, New York State had a total of 194 tornadoes that measured 1 or greater on the Fujita scale. Let the random variable x represent the number of such tornadoes to hit New York State in one year, and assume that it has a Poisson distribution. What is the mean number of such New York tornadoes in one year? What is the standard deviation? What is the variance?

Problem 2CQQ There are 100 questions from an SAT test, and they are all multiple choice with possible answers of a, b, c, d, e. For each question, only one answer is correct. Find the mean number of correct answers for those who make random guesses for all 100 questions.

Ohio Pick 4 In Ohio's Pick 4 game, you pay $1 to select a sequence of four digits, such as 7709 . If you buy only one ticket and win, your prize is $5000 and your net gain is $4999. a. If you buy one ticket, what is the probability of winning? b. Construct a table describing the probability distribution corresponding to the purchase of one Pick 4 ticket. c. If you play this game once every day, find the mean number of wins in years with exactly 365 days. d. If you play this game once every day, find the probability of winning exactly once in 365 days. e. Find the expected value for the purchase of one ticket.

Problem 2RE In Exercises, assume that 40% of the population has brown eyes (based on data from Dr. P. Sorita at Indiana University). Brown Eyes Find the probability that among six randomly selected people, exactly four of them have brown eyes.

Problem 3BSC Poission Approximation to Binomial Assume that we want to find the probability of getting at least one win when playing the Texas Pick 3 lottery 50 times. For one bet, there is a 1 /1000 probability of winning. If we want to use the Poisson distribution as an approximation to the binomial, are the requirements satisfied? Why or why not?

Problem 3CQQ Using the same SAT questions described in Exercise, find the standard deviation for the numbers of correct answers for those who make random guesses for all 100 questions. Exercise There are 100 questions from an SAT test, and they are all multiple choice with possible answers of a, b, c, d, e. For each question, only one answer is correct. Find the mean number of correct answers for those who make random guesses for all 100 questions.

Problem 3CRE Tennis Challenge In the last U.S. Open tennis tournament, there were 611 challenges made by singles players, and 172 of them resulted in referee calls that were overturned. The accompanying table lists the results by gender. Challenge Upheld with Overturned Call Challenge Rejected with No Change Challenges by Men 121 279 Challenges by Women 51 160 a. If one of the 611 challenges is randomly selected, what is the probability that it resulted in an overturned call? ________________ b. If one of the challenges made by the men is randomly selected, what is the probability that it resulted in an overturned call? ________________ c. If one of the challenges made by the women is randomly selected, what is the probability that it resulted in an overturned call? ________________ d. If one of the overturned calls is randomly selected, what is the probability that the challenge was made by a woman? ________________ e. If two different challenges are randomly selected with replacement, find the probability that they both resulted in an overturned call. ________________ f. If one of the 611 challenges is randomly selected, find the probability that it was made by a man or was upheld with an overturned call. ________________ g. If one of the challenged calls is randomly selected, find the probability that it was made by a man given that the call was upheld with an overturned call.

Problem 3RE In Exercises, assume that 40% of the population has brown eyes (based on data from Dr. P. Sorita at Indiana University). Brown Eyes Groups of 600 people are randomly selected. Find the mean and standard deviation for the numbers of people with brown eyes in such groups, then use the range rule of thumb to identify the range of usual values for those with brown eyes. For such a group of 600 randomly selected people, is 200 with brown eyes unusually low or high?

Problem 4BSC Poisson Approximation to Binomial Assume that we plan to play the Texas Pick 3 lottery 100 times. For one bet, there is a 1 /1000 probability of winning. If we want to use the Poisson distribution as an approximation to the binomial, are the requirements satisfied? If we use the Poisson distribution to find the probability of 101 wins, we get an extremely small positive number, so is it correct to conclude that the probability of 101 wins is possible, but highly unlikely?

Problem 4CQQ If boys and girls are equally likely, groups of 400 births have a mean of 200 girls and a standard deviation of 10 girls. Is 232 girls in 400 births an unusually high number of girls?

Gender Gap In recent years, the discrepancy between incomes of women and men has been shrinking, but it continues to exist. The accompanying graph illustrates the current gap. The graph shows that for every $100 earned by men, women earn $82.80 (based on data from the Bureau of Labor Statistics). Identify what is wrong with the graph, then redraw it so that it is not deceptive.

Problem 4RE In Exercises, assume that 40% of the population has brown eyes (based on data from Dr. P. Sorita at Indiana University). Brown Eyes When randomly selecting 600 people, the probability of exactly 239 people with brown eyes is P(239) = 0.0331. Also, P(239 or fewer) = 0.484. Which of those two probabilities is relevant for determining whether 239 is an unusually low number of people with brown eyes? Is 239 an unusually low number of people with brown eyes?

Problem 5CRE Random Digits The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are randomly selected for applications including the selection of lottery numbers and the selection of telephone numbers to be called as part of a survey. In the following tables, the table at the left summarizes actual results from 100 randomly selected digits, and the table at the right summarizes the probabilities of the different digits. Digit Frequency 0 9 1 7 2 12 3 10 4 10 5 11 6 8 7 8 8 14 9 11 Digit x P(x) 0 0.1 1 0.1 2 0.1 3 0.1 4 0.1 5 0.1 6 0.1 7 0.1 8 0.1 9 0.1 a. What is the table at the left called? ________________ b. What is the table at the right called? ________________ c. Use the table at the left to find the mean. Is the result a statistic or a parameter? ________________ d. Use the table at the right to find the mean. Is the result a statistic or a parameter? ________________ e. If you were to randomly generate 1000 such digits, would you expect the mean of these 1000 digits to be close to the result from part (c) or part (d)? Why?

Problem 5BSC Aircraft Accidents. Assume that the Poisson distribution applies, assume that the mean number of aircraft accidents in the United States is8.5per month (based on data from the National Transportation and Safety Board), and proceed to find the indicated probability. Find P(0), the probability that in a month, there will be no aircraft accidents. Is it unlikely to have a month with no aircraft accidents?

Problem 6BSC Aircraft Accidents. Assume that the Poisson distribution applies, assume that the mn number of aircraft accidents in the United States is8.5 per month (based on data from the National Transportation and Safety Board), and proceed to find the indicated probability. Find P(6), the probability that in a month, there will be 6 aircraft accidents. Is it unlikely to have a month with exactly 6 aircraft accidents?

Problem 6CRE Investing in College Based on a USA Today poll, assume that 10% of the population believes that college is no longer a good investment. a. Find the probability that among 16 randomly selected people, exactly 4 believe that college is no longer a good investment. ________________ b. Find the probability that among 16 randomly selected people, at least 1 believes that college is no longer a good investment. ________________ c. The poll results were obtained by Internet users logged on to the USA Today Web site, and the Internet users decided whether to ignore the posted survey or respond. What type of sample is this? What does it suggest about the validity of the results?

Problem 7BSC Aircraft Accidents. Assume that the Poisson distribution applies, assume that the mean number of aircraft accidents in the United States is8.5per month (based on data from the National Transportation and Safety Board), and proceed to find the indicated probability. Find P(10), the probability that in a month, there will be 10 aircraft accidents. Is it unlikely to have a month with exactly 10 aircraft accidents?

Problem 7RE Brand Recognition In a study of brand recognition of the Kindle eReader, four consumers are interviewed. If x is the number of consumers in the group who recognize the Kindle brand name, then x can be 0, 1, 2, 3, or 4. If the corresponding probabilities are 0.026, 0.154, 0.346, 0.246, and 0.130, does the given information describe a probability distribution? Why or why not? x P(x) 0 0.674 1 0.280 2 0.044 3 0.003 4 0+

Problem 8RE Expected Value for Deal or No Deal In the television game show Deal or No Deal, contestant Elna Hindler had to choose between acceptance of an offer of $193,000 or continuing the game. If she continued to refuse all further offers, she would have won one of these five equally likely prizes: $75, $300, $75,000, $500,000, and $1,000,000. Find her expected value if she continued the game and refused all further offers. Based on the result, should she accept the offer of $193,000, or should she continue?

Problem 8BSC Aircraft Accidents. Assume that the Poisson distribution applies, assume that the mean number of aircraft accidents in the United States is8.5per month (based on data from the National Transportation and Safety Board), and proceed to find the indicated probability. Find P(12), the probability that in a month, there will be 12 aircraft accidents. Is it unlikely to have a month with exactly 12 aircraft accidents?

Problem 9BSC Use the Poisson distribution to find the indicated probabilities. Earthquakes in the United States Various sources provide different earthquake data, but assume that for a recent 4l-year period in the United States, there were 268 earthquakes measured at 6.0 or higher on the Richter scale (based on U.S. Geological Survey data). a. Find the mean number of earthquakes per year. b. Find the probability that in a given year, there is at least one earthquake in the United States that measures 6.0 or higher on the Richter scale. c. Is it unlikely to have a year without any earthquakes that measure 6.0 or higher on the Richter scale? Why or why not?

Problem 9CQQ In Exercises, use the following: Five U.S. domestic flights are randomly selected, and the table in the margin lists the probabilities for the number that arrive on time (based on data from the Department of Transportation). Assume that five flights are randomly selected. Is 0 an unusually low number of flights arriving on time? x P(x) 0 0+ 1 0.006 2 0.048 3 0.198 4 0.409 5 0.338

Problem 9RE Expected Value for a Magazine Sweepstakes Reader’s Digest ran a sweepstakes in which prizes were listed along with the chances of winning: $1,000,000 (1 chance in 90,000,000), $100,000 (1 chance in 110,000,000), $25,000 (1 chance in 110,000,000), $5,000 (1 chance in 36,667,000), and $2,500 (1 chance in 27,500,000). a. Assuming that there is no cost of entering the sweepstakes, find the expected value of the amount won for one entry. b. Find the expected value if the cost of entering this sweepstakes is the cost of a postage stamp. Is it worth entering this contest?

Problem 10BSC Use the Poisson distribution to find the indicated probabilities. Earthquakes in the World Various sources provide different earthquake data, but assume that for a recent 41-year period in the world, there were 5469 earthquakes measured at 6.0 or higher on the Richter scale (based on U.S. Geological Survey data). a. Find the mean number of earthquakes per year. b. Find the probability >that in a given year, there are exactly 133 earthquakes that measures 6.0 or higher on the Richter scale. c. If, in a particular year, there are exactly 133 earthquakes that measure 6.0 or higher on the Richter scale, would it make sense to report that this is a year with an unusual number of earthquakes? Why or why not?

Problem 10RE Phone Calls In the month preceding the creation of this exercise, the author made 18 phone calls in 30 days. No calls were made on 19 days, 1 call was made on 8 days, and 2 calls were made on 5 days. a. Find the mean number of calls per day. b. Use the Poisson distribution to find the probability of no calls in a day. c. Based on the probability in part (b), how many of the 30 days are expected to have no calls? ________________ d. There were actually 18 days with no calls. How does this actual result compare to the expected value from part (c)?

Use the Poisson distribution to find the indicated probabilities. Radioactive Decay Radioactive atoms are unstable because they have too much energy. When they release their extra energy, they are said to decay. When studying cesium­137, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 977,287 radioactive atoms, so 22,713 atoms decayed during 365 days. a. Find the mean number of radioactive atoms that decayed in a day. b. Find the probability that on a given day, 50 radioactive atoms decayed.

Problem 12BSC Use the Poisson distribution to find the indicated probabilities. Deaths from Horse Kicks A classical example of the Poisson distribution involves the number of deaths caused by horse kicks to men in the Prussian Army between 1875 and 1894. Data for 14 corps were combined for the 20-year period, and the 280 corps-years included a total of 196 deaths. After finding the mean number of deaths per corps-year, find the probability that a randomly selected corps-year has the following numbers of deaths: (a) 0, (b) 1, (c) 2, (d) 3, (e) 4. The actual results consisted of these frequencies: 0 deaths (in 144 corps-years) ;1 death (in 91 corps-years); 2 deaths (in 32 corps-years); 3 deaths (in 11 corps-years); 4 deaths (in 2 corps-years). Compare the actual results to those expected by using the Poisson probabilities. Does the Poisson distribution serve as a good tool for predicting the actual results?

Problem 13BSC Use the Poisson distribution to find the indicated probabilities. World War II Bombs In Exercise we noted that in analyzing hits by V-l buzz bombs in World War II, South London was partitioned into 576 regions, each with an area of 0.25 km2. A total of 535 bombs hit the combined area of 576 regions. a. Find the probability that a randomly selected region had exactly 2 hits. b. Among the 576 regions, find the expected number of regions with exactly 2 hits. c. How does the result from part (b) compare to this actual result: There were 93 regions that had exactly 2 hits? Exercise Notation In analyzing hits by V-l buzz bombs in World War II, South London was partitioned into 576 regions, each with an area of 0.25 km2. A total of 535 bombs hit the combined area of 576 regions. Assume that we want to find the probability that a randomly selected region had exactly two hits. In applying Formula 5-9, identify the values of µ, x, and e. Also, briefly describe what each of those symbols represents.

Problem 14BSC Use the Poisson distribution to find the indicated probabilities. Disease Cluster Neuroblastoma, a rare form of malignant tumor, occurs in 11 children in a million, so its probability is 0.000011. Four cases of neuroblastoma occurred in Oak Park, Illinois, which had 12,429 children. a. Assuming that neuroblastoma occurs as usual, find the mean number of cases in groups of 12,429 children. b. Using the unrounded mean from part (a), find the probability that the number of neuroblastoma cases in a group of 12,429 children is 0 or 1. c. What is the probability of more than one case of neuroblastoma? d. Does the cluster of four cases appear to be attributable to random chance? Why or why not?

Problem 15BSC Use the Poisson distribution to find the indicated probabilities. Chocolate Chip Cookies In the production of chocolate chip cookies, we can consider each cookie to be the specified interval unit required for a Poisson distribution, and we can consider the variable x to be the number of chocolate chips in a cookie. Table 3-1 is included with the Chapter Problem for Chapter 3, and it includes the numbers of chocolate chips in 34 different Keebler cookies. The Poisson distribution requires a value for ? so use 30.4, which is the mean number of chocolate chips in the 34 Keebler cookies. Assume that the Poisson distribution applies. a. Find the probability that a cookie will have 26 chocolate chips, then find the expected number of cookies with 26 chocolate chips among 34 different Keebler cookies, then compare the result to the actual number of Keebler cookies with 26 chocolate chips. b. Find the probability that a cookie will have 30 chocolate chips, then find the expected number of cookies with 30 chocolate chips among 34 different Keebler cookies, then compare the result to the actual number of Keebler cookies with 30 chocolate chips.

Problem 17BB Poisson Approximation to Binomial Distribution An experiment consists of rolling a single die 12 times and the variable x is the number of times that the outcome is 6. a. Can the Poisson distribution be used to find the probability that the outcome of 6 occurs exactly 3 times? Why or why not? ________________ b. If the Poisson distribution is used, is the result OK? Why or why not?

Problem 6RE In Exercises , refer to the table in the margin. The random variable x is the number of males with tinnitus (ringing ears) among four randomly selected males (based on data from “Prevalence and Characteristics of Tinnitus among US Adults” by Shargorodsky et al,American Journal of Medicine, Vol. 123, No. 8). Tinnitus Find the mean and standard deviation for the random variable x. Use the range rule of thumb to identify the range of usual values for the number of males with tinnitus among four randomly selected males. Is it unusual to get three males with tinnitus among four randomly selected males?

Problem 7CQQ In Exercises, use the following: Five U.S. domestic flights are randomly selected, and the table in the margin lists the probabilities for the number that arrive on time (based on data from the Department of Transportation). Assume that five flights are randomly selected. What does the probability of 0+ indicate? Does it indicate that among five randomly selected flights, it is impossible that none of them arrives on time? x P(x) 0 0+ 1 0.006 2 0.048 3 0.198 4 0.409 5 0.338

Problem 16BSC Use the Poisson distribution to find the indicated probabilities. Chocolate Chip Cookies Consider an individual chocolate chip cookie to be the specified interval unit required for a Poisson distribution, and consider the variable x to be the number of chocolate chips in a cookie. Table 3-1 is included with the Chapter Problem for Chapter 3, and it includes the numbers of chocolate chips in 40 different reduced fat Chips Alroy cookies. The Poisson distribution requires a value for µ, so use 19.6, which is the mean number of chocolate chips in the 40 reduced fat Chips Ahoy cookies. Assume that the Poisson distribution applies. a. Find the probability that a cookie will have 18 chocolate chips, then find the expected number of cookies with 18 chocolate chips among 40 different reduced fat Chips Ahoy cookies, then compare the result to the actual number of reduced fat Chips Ahoy cookies with 18 chocolate chips. b. Find the probability that a cookie will have 21 chocolate chips, then find the expected number of cookies with 21 chocolate chips among 40 different reduced fat Chips Ahoy cookies, then compare the result to the actual number of reduced fat Chips Ahoy cookies with 21 chocolate chips.

Problem 8CQQ In Exercises, use the following: Five U.S. domestic flights are randomly selected, and the table in the margin lists the probabilities for the number that arrive on time (based on data from the Department of Transportation). Assume that five flights are randomly selected. What is the probability that at least three of the five flights arrive on time? x P(x) 0 0+ 1 0.006 2 0.048 3 0.198 4 0.409 5 0.338

Problem 5CQQ If boys and girls are equally likely, groups of 400 births have a mean of 200 girls and a standard deviation of 10 girls. Is 185 girls in 400 births an unusually low number of girls?

Problem 5RE In Exercises , refer to the table in the margin. The random variable x is the number of males with tinnitus (ringing ears) among four randomly selected males (based on data from “Prevalence and Characteristics of Tinnitus among US Adults” by Shargorodsky et al,American Journal of Medicine, Vol. 123, No. 8). Tinnitus Does the table describe a probability distribution? Why or why not?

Problem 6CQQ In Exercises, use the following: Five U.S. domestic flights are randomly selected, and the table in the margin lists the probabilities for the number that arrive on time (based on data from the Department of Transportation). Assume that five flights are randomly selected. Does the table describe a probability distribution? x P(x) 0 0+ 1 0.006 2 0.048 3 0.198 4 0.409 5 0.338

Problem 10CQQ In Exercises, use the following: Five U.S. domestic flights are randomly selected, and the table in the margin lists the probabilities for the number that arrive on time (based on data from the Department of Transportation). Assume that five flights are randomly selected. Is 5 an unusually high number of flights arriving on time? x P(x) 0 0+ 1 0.006 2 0.048 3 0.198 4 0.409 5 0.338