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CLASS SOME ACTIVITIES FOR CHAPTER 7 Activity 1 A Door Prize A party host gives a door prize to one guess chosen at random. There are 48 men and 42 women at the party. What is the probability that the prize goes to a woman? Explain how you arrived at your answer. (.4667) Activity 2 Land in Canada Canada's national statistics agency, statistics Canada, says that the land area of Canada is 9,094,000 square kilometers. Of this land, 4,176,000 square kilometers are forested. Choose a square kilometer of land in Canada at random. (a) What is the probability that the area you choose is forested. (b) What is the probability that it is not forested? (.5408) (.4592) Activity 3 Car Colors Choose a new car or light truck at random and note its color. Here are the probabilities of the most popular colors for vehicles made in North America in 2005. Color Probability Silver White 0.18 0.17 Gray Blue 0.15 0.12 Black Red 0.11 0.11 (a) What is the probability that the vehicle you choose has any color other than the six listed? (.16) (b) What is the probability that a randomly chosen vehicle is neither silver nor white? (.65) 1 Activity 4 - Genetics A gene is composed of two alleles. An allele can be either dominant or recessive. Suppose a husband and wife, who are both carriers of the sickle-cell anemia allele but do not have the disease, decide to have a child. Because both parents are carriers of the disease, each has one dominant normal-cell allele and one recessive sickle-cell allele. Therefore, the genotype of each parent is Ss. Each parent contributes one allele to his offspring, with each allele being equally likely. (a) List the possible genotypes of their offspring. (b) What is the probability that the offspring will have sickle-cell anemia? In other words, what is the probability the offspring will have genotype ss? Interpret the probability. (c) What is the probability that the offspring will not have sickle-cell anemia but will be a career? In other words, what is the probability that the offspring will have one dominant normal-cell allele and one recessive sickle-cell allele? Interpret this probability. Activity 5 - Birthdays Exclude leap years from the following calculations and assume each birthday is equally likely: (a) Determine the probability that a randomly selected person has a birthday on the 1st day of a month. Interpret this probability. (12/365) (b) Determine the probability that a randomly selected person has a birthday on the 31st day of a month. Interpret this probability. (7/365) (c) Determine the probability that a randomly selected person was born in December. Interpret this probability. (31/365) (d) Determine the probability that a randomly selected person has a birthday on November 8. Interpret this probability. (1/365) (e) If you just met somebody and she asked you to guess her birthday, are you likely to be correct? 1/365 - No, it is unlikely (f ) Do you think it is appropriate to use the methods of classical probability to compute the probability that a person is born in December? (No) The 7 days of the week are actually not equally likely. Less people are born on Saturday and Sunday. 2 Activity 6 - Guessing Yet Passing a Pop Quiz For a three-question multiple choice pop quiz, a student is totally unprepared and randomly guesses the answer to each question. (a) If each question has 2 options, then what is the probabilities of the correct answer for any given question? Explain. (1/2) (a) Find the probabilities of each possible student outcomes for the quiz, in terms of whether each response is correct (C) or incorrect (I). (1/8) (b) Find the probability the student passes, answering at least two correctly. (1/2) Activity 7 - What Are the Chances of a Taxpayer Being Audited? April 15 is the tax day in the US - the deadline for filing federal income tax forms. The main factor in the amount of tax owed is a taxpayer's income level. Each year, the IRS audits a sample of tax forms to verify their accuracy. The following table is a contingency table that cross-tabulates the 80.2 million long-form federal returns received in 2002 by the taxpayer's income level and whether the tax form was audited. Table 1: Contingency Table Cross-Tabulating Tax Forms by Income Level and Whether Audited reported in thousands. Income Level Under $25,000 $25,000 - $49,000 $50,000 - $99,000 $100,000 or more Total Whether Yes 90 71 60 80 310 Audited No 14010 30629 24631 10620 79890 Total 14100 30700 24700 10700 80200 (a) What is the sample space? (Under $25,000, Yes), (Under $25,000, No),(Under $25,000-$49,999, Yes), and so forth. (b) For a randomly chosen taxpayer in 2002, what is the probability that a taxpayer is audited and has income of $100,000 or more? (0.0997) (c) For a randomly chosen taxpayer in 2002, what is the probability of an audit? (0.0039) 3 (d) For a randomly chosen taxpayer in 2002, what is the probability of an income of $100,000 or more? (0.1334) (e) For a randomly chosen taxpayer in 2002, what is the probability of an income less $100,000? (0.8667) (e) Given that a taxpayer has an income of $100,000 or more, what is the probability of him being audited? (0.007) (f ) Given that a taxpayer was audited, what is the probability of him having an income of $100,000 or more? (0.2581) Activity 8 - Marital Status Consider the data shown below which represent the marital status of males and females 18 years old or older in the US in 2003. Males (in millions) 28.6 62.1 2.7 9.0 102.4 Females Total (in millions) (in millions) 23.3 51.9 62.8 124.9 11.3 14.0 12.7 21.7 110.1 212.5 Never married Married Widowed Divorced Total (in millions) (a) Determine the probability that a randomly selected US resident 18 years old or older is male. (0.482) (b) Determine the probability that a randomly selected US resident 18 years old or older is widowed. (0.066) (c) Determine the probability that a randomly selected US resident 18 years old or older is widowed or divorced. (0.168) (d) Determine the probability that a randomly selected US resident 18 years old or older is male or widowed. (0.535) (e) Compute the probability that a randomly selected male has never married. conditional probability - (0.279) - There is a 27.9% probability that the randomly selected individual has never married, given that he is a male (f ) Compute the probability that a randomly selected individual who has never married is male. (0.551) - There is a 55.1% probability that the randomly selected is a male, given that he or she has never married 4 Activity 9 - Should We Be Surprised by Finding No Women on a Jury? A jury of 12 people is chosen for a trial. The defense attorney claims it must have been chosen in a biased manner, because 50% of the city's adult residents are female yet the jury contains no women. If the jury were randomly chosen from the population, (a) what is the probability the jury would have no female? (1/4096=0.00024) (.99976) (b) what is the probability the jury would have at least one female? Activity 10 (a) A small community consists of 10 men, each of whom has 3 sons. If one man and one of his sons are to be chosen as father and son of the year, how many choices are possible? ( 10 3 = 30) 1 1 (b) There are 5 routes available between A and B, 4 between B and C, and 7 between C and D. What is the total number of available routes between A and D? (5 4 7 = 140) (c) A college committee consists of 3 freshman, 4 sophomers, 5 juniors, and 2 seniors. A subcommittee of 4, consisting of 1 individual from each class, is to be chosen. How many different subcommittees are possible? ( 3 4 5 2 = 120) 1 1 1 1 (d) Suppose a local area network requires eight characters for user names. Lower- and uppercase letters are considered the same. How many user names are possible for the local area network? (268 ) (e) Suppose a local area network requires eight characters for password. The first character must be a letter, but the remaining seven characters can either be a letter or a digit (0 through 9). Lower- and uppercase letters are considered the same. How many user names are possible for the local area network? (26 367 ) (f ) A combination lock has 50 numbers on it. To open it, you turn counterclockwise to a number, then rotate clockwise to a second number, and then counterclockwise to the third number. How many different lock combinations are there? What is the probability of guessing a lock combination on the first try? (503 - 1/503 ) 5 Activity 11 - Guessing Yet Passing a Pop Quiz For a four-question multiple choice pop quiz, a student is totally unprepared and randomly guesses the answer to each question. If each question has five options, then the probability of selecting the correct answer for any given question is 1/5, or 0.20. With guessing, the response on one question is not influenced by the response on another question. Thus, whether one question is answered correctly is independent of whether or not another question is answered correctly. (a) Find the probabilities each of possible student outcomes for the quiz, in terms of whether each response is correct (C) or incorrect (I). (b) Find the probability the student passes, answering at least three question correctly. (0.0272) Activity 12 Suppose two doctors, A and B, test all patients coming into a VD clinic for syphilis. Let the events A+ = {doctor A make a positive diagnosis}, + B = {doctor B make a positive diagnosis}. Suppose that doctor A diagnoses 10% of all patients as positive, doctor B diagnoses 17% of all patients as positive, and both doctors diagnose 8% of all patients as positive. (a) Are the events A+ and B + independent? Comment! (No) - This is expect because there should be similarities between how two doctors diagnose patients with syphilis. (b) Suppose that a patient is referred for further lab tests if either doctor A or B makes a positive diagnosis. What is the probability that a patient will be referred for further lab tests? (0.19) (c) Find conditional probability that doctor B makes a positive diagnosis of syphilis given that doctor A makes a positive diagnosis. (.8) - Thus doctor B will confirm doctor A's positive diagnosis 80% of the time (d) What is the conditional probability that doctor B makes a positive diagnosis of syphilis given that doctor A makes a negative diagnosis? (.10) Thus, when A doctor diagnoses a patient as negative, doctor B will contradict the diagnosis 10% of the time 6 Activity 13 70% of the light aircrafts that disappear while in flight in a certain country are subsequently discovered. Of the aircraft are discovered, 60% have an emergency locator, whereas 90% of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared. (a) If it has an emergency locator, what is the probability that it will not be discovered? (.067) (b) If it does not have an emergency locator, what is the probability that it will be discovered? (.509) Activity 14 A paint-store chain produces an sells latex and semigloss paint. Based on longrange sales, the probability that a customer will purchase latex is 0.75. Of those that purchase latex paint, 60% also purchase rollers. But 30% of semigloss buyers purchase rollers. A randomly selected buyer purchases a roller and a can of paint. What is the probability that the paint is latex? (.8571) - conditional probability - use 2 2 table Activity 15 At a certain gas station, 40% of the customers use regular unleaded gas, 35% use extra unleaded gas, and 25% use premium unleaded gas. Of those customers using regular gas, only 30% fill their tanks. Of those customers using extra gas, 60% fill their tanks, whereas of those using premium, 50% fill their tanks. (a) What is the probability that the next customer will request extra unleaded gas and fill the tank? (.21) (b) What is the probability that the next customer fills the tank? (0.455) (c) If the next customer fills the tank, what is the probability that extra unleaded gas is requested? (.4615) (d) If the next customer fills the tank, what is the probability that regular gas is requested? (.2637) (e) If the next customer fills the tank, what is the probability that premium gas is requested? (.2747) 7 Activity 16 - Life Expectancy The probability that a randomly selected 40-year-old male will live to be 41 years old is 0.99718 according to the National Vital Statistics Report, Vol. 48, No.18. (a) What is the probability that two randomly selected 40-year-old males will live to be 41 years old? (.997182 =99.44%) - use independence (b) What is the probability that five randomly selected 40-year-old males will live to be 41 years old? (.997185 = 98.60%) - use independence (c) What is the probability that a least one of the five randomly selected 40-yearold males will not live to be 41 years old? Would it be unusual that at least one of five randomly selected 40-year-old males will not live to be 41 years old? (0.014) - yes, it is unusual Activity 17 - Defense System Suppose a satellite defense system is established in which four satellites acting independently have a 0.9 probability of detecting an incoming ballistic missile. What is the probability at least one the four satellites detects an incoming ballistic missile? Would you feel safe with such a system? (1 - .14 =.9999%) Activity 18 - E.P.T. Pregnancy Tests The packaging of an E.P.T. Pregnancy Test states that the test is "99% accurate at detecting typical pregnancy hormone levels." Assume the probability that a test will correctly identify a pregnancy is 0.99. Suppose 12 randomly selected pregnant women with typical hormone levels are each given the test. (a) What is the probability that all 12 tests will be positive? - use independence (.9912 =0.8864) (0.1136) (b) What is the probability that at least one test will not be positive? 8 Activity 19 - Driver Fatalities The following data represent the number of driver fatalities in the US in 2002 by age for male and female drivers: Age Under 16 16-20 21-34 35-54 55-69 70 or older Male 228 5696 13,553 14,395 4937 3159 Female 108 2386 4148 5017 1708 1529 (a) What is the probability that a randomly selected driver fatality who was male was 16 to 20 years old? (0.136) (b) What is the probability that a randomly selected driver fatality who was 16 to 20 years old was male? (0.705) (c) Suppose you are a police officer called to the scene of a traffic accident...