5. discrete exercises

Introductory Statistics Explained (1.11) Exercises Discrete Random Variables and Discrete Probability Distributions © 2023, 2024 Jeremy Balka Chapter 4: Discrete Random Variables v1.11 W24 Draft J.B.’s strongly suggested exercises: 2 , 5 , 6 , 7 , 8 , 10 , 13 , 14 , 15 , 19 , 20 , 22 , 23 , 25 , 26 , 28 , 30 , 48 , 51 , 53 , 54 , 57 , 58 , 62 , 64 , 66 (a-d). NB The section titles and numbers are not yet synced up with the text. 1 Introduction 2 Discrete and Continuous Random Variables 1. For each of the following random variables, state whether they are discrete or continuous. (a) The number of poker hands dealt in a casino in an hour. (b) The time until completion of a randomly selected poker hand in a casino. (c) The amount of cheese on a randomly selected cheeseburger at a fast food restaurant. (d) The number of moves in a game at the World Chess Championship. (e) The duration of a match at the World Chess Championship. (f) The number of die rolls required to roll a six 18 times in a row. (g) The sum of the numbers that come up on the top face when a die is rolled 400 times, divided by 3617. 3 Discrete Probability Distributions 2. Suppose that two cards are drawn without replacement from a standard 52 card deck. Let X represent the number of hearts drawn. Find the probability distribution of X . 1 x heart anwn P X 0 38 IE EI f Hf I 2 hearts dont 0 non hearts Pl 2 mn heart prob 3 52 non heart prob 39 52

3. Consider the following probability distribution of a random variable X . x 10 20 30 40 p ( x ) 0.2 0.2 0.5 ? (a) Is X discrete or continuous? (b) What is the value of the missing probability? (c) What is the most likely value of X ? (d) What is P ( X < 32) ? (e) What is the conditional probability X is less than 25, given X is less than 35? 4. Which of the following are valid discrete probability distributions? (a) x - 10 1.7 30000 p ( x ) 0.2 0.2 0.6 (b) x 1 2 3 p ( x ) 0.2 0.2 0.2 (c) x 0 . 1 0.2 0.3 4000 500000 p ( x ) 0.2 0.2 0.2 0.2 0.2 (d) x 1 2 3 4 5 p ( x ) - 0 . 2 0.3 0.3 0.4 0.2 3.1 The Expectation and Variance of Discrete Random Variables 3.1.1 Calculating the expected value and variance of a discrete random variable 5. Consider the following probability distribution of a random variable X . x - 2 . 5 0 2.5 5.0 7.5 p ( x ) 0.10 0.10 0.10 0.40 ? (a) Is X discrete or continuous? (b) What is the value of the missing probability? (c) What is the most likely value of X ? (d) What is P ( - 1 < X < 4) ? (e) What is the expected value of X ? (f) What is the standard deviation of X ? 6. Hellin’s law is a rough guideline that gives approximate probabilities of multiple births in preg- nancies that are not the result of fertility treatments. Hellin’s law states that approximately 1 in 89 pregnancies result in twins, approximately 1 in 89 2 pregnancies result in triplets, approximately 1 in 89 3 pregnancies result in quadruplets, and so on. Not done 0.18 0.16 o.isto 16 0.12 gg f x.no 5 2251 35 239 AND 53 in go in L 010.10.1 0.420.1 0.3 is missing KIIII.si ci ni i ii i i as.nusnn in u.as mn iiiiiiiiii i

(a) What is the mean time to completion of the entire procedure? (b) What is the standard deviation of the time to completion of the entire procedure? (c) Suppose we wish to measure time in hours instead of minutes. What are the mean, variance, and standard deviation of the time (in hours) to completion? 4 The Bernoulli Distribution 11. Suppose a single card is drawn from a well-shu ffl ed 52 standard deck. (a) What is the distribution of the number of eights that are drawn? (b) What is the mean number of eights that are drawn? (c) What is the standard deviation of the number of eights that are drawn? 5 The Binomial Distribution 12. Suppose X is a binomial random variable with parameters n = 15 and p = 0 . 2 . (a) What is P ( X = 3) ? (b) What is P ( X  3) ? (c) What is P ( X > 3) ? (d) What is P (2  X < 5) ? (e) What is P ( X = 3 | X  4) (f) What is the mean of X ? (g) What is the standard deviation of X ? 13. According to the United States Centers for Disease Control and Prevention, approximately 3% of babies in the United States are born with major structural or genetic birth defects. Suppose 50 newborns in the United States are randomly selected. (a) What is the probability that exactly 2 have major structural or genetic birth defects? (b) What is the probability that no more than 2 have major structural or genetic birth defects? (c) What is the probability that exactly 10 have major structural or genetic birth defects? (d) What is the expectation of the number with major structural or genetic birth defects? (e) What is the standard deviation of the number with major structural or genetic birth defects? 14. Which of the following statements are true? (a) The variance of a binomial distribution is always greater than the mean. (b) A binomial random variable can take on negative values. (c) The mean of a binomial random variable can be negative. (d) The variance of a binomial random variable can be negative. (e) A binomial random variable is a discrete random variable. (f) Every discrete random variable is a binomial random variable. (g) A binomial random variable represents a count. (h) The mean of a binomial random variable is np . EXT 148min Var 02 T 235.44mins sD a It 15.34min Yfits fef stay same 6 125 117 148 no 2 05 15.34M toELD fox148 2 u6hrso2 toavarlt 23s.ua 0.0654 o foxsdltl foxls.sn 0.256min abinoml2 so.o.us 0.25s iiiiiiii.it i i.i I must be at Fes must be less equal to mean in F I n np

(i) The standard deviation of a binomial random variable is p np (1 - p ) . (j) For any given n , the variance of a binomial random variable is greatest when p = 0 . 5 . 5.1 Binomial or Not? 15. Which of the following random variables have a binomial distribution? (a) The number of hearts if 10 cards are drawn without replacement from a standard deck. (b) The number of times heads comes up if a biased coin is tossed 10 times. (c) The number of putts a golfer holes in their next 10 attempts. (d) The number of newborn babies in a maternity ward on a randomly selected day. (e) The amount of money withdrawn by a randomly selected customer at a bank. 16. Which of the following random variables have a binomial distribution? (a) The number of hearts if 10 cards are drawn with replacement from a standard deck. (b) The number of fingers on a randomly selected newborn. (c) The number of times a sum of 7 is rolled when a pair of dice is rolled 14 times. (d) The number of children a randomly selected woman has. (e) The number of times a randomly selected person uses the washroom on a randomly selected day. (f) The weight of 15 randomly selected playing cards. 5.2 A Binomial Example with Probability Calculations 17. A friend of yours says they have extra-sensory perception (ESP), and claims they can pull the king of spades from a well shu ffl ed 52-card deck approximately 20% of the time. You think they are being foolish and you decide to test their claim. You open a brand new deck of cards, shu ffl e it well, and ask them to draw the king of spades. You repeat this procedure until they have drawn a card 50 times. They manage to draw the king of spades twice in the 50 draws. (a) If your friend does not have ESP, and is merely randomly pulling a card each time, what is the probability they draw the king of spades at least twice in 50 draws? (b) Does this experiment give strong evidence that your friend does not have ESP? 6 The Hypergeometric Distribution 18. An urn contains 22 white and 18 red balls. Six balls are randomly chosen without replacement. (a) What is the probability that exactly 4 white balls are chosen? (b) What is the probability that no more than 1 white ball is chosen? (c) What is the mean number of white balls chosen? What is the mean number of red balls chosen? 19. You have an idea to start a business that delivers pizzas using stretch limousines. You need to hire 5 drivers, and 20 people apply for the positions. Unknown to you, 7 of these 20 people have criminal records. Suppose you (foolishly) decide to hire a random sample of 5 drivers from the 20 applicants. not e yup P grotest www i E EEI iiiiii x.io n t Pirime 7120 Protrime 13 20

(e) If X has a Poisson distribution, then P ( X = 0) < P ( X = 1) . (f) Every random variable that represents a count is a Poisson random variable. 7.2 The Relationship Between the Poisson and Binomial Distributions 24. In which one of the following situations would the Poisson distribution provide the most reasonable approximation to the binomial distribution? (a) X has a binomial distribution with n = 5 and p = 0 . 5 . (b) X has a binomial distribution with n = 10 , 000 and p = 0 . 9 . (c) X has a binomial distribution with n = 10 and p = 0 . 2 . (d) X has a binomial distribution with n = 500 and p = 0 . 01 . (e) X has a binomial distribution with n = 5 and p = 0 . 001 . 25. Protanopia is a kind of colour blindness that a ff ects approximately 1% of males (and a much smaller percentage of females). Individuals with protanopia do not perceive red light normally, and have di ffi culty distinguishing between red and green and between red and blue. Suppose 100 males are randomly selected from a large population in which 1% of the males have protanopia. (a) Using the binomial probability mass function, find the probability that exactly 2 of these 100 males have protanopia. Also find the approximate probability based on the Poisson distribution. (b) Using the binomial probability mass function, find the probability that no more than 2 of these 100 males have protanopia. Also find the approximate probability based on the Poisson distribution. (c) Based on the binomial distribution, what is the variance of the number of males with protanopia? What is the variance based on the Poisson approximation? 7.3 Poisson or Not? More Discussion on When a Random Variable has a Poisson distribution 26. Suppose it is known that the number of students entering a University Centre averages 4.5 per minute between noon and 12:30 pm. Would the number of students that enter this University Centre in a randomly selected minute in this time frame follows a Poisson distribution with λ = 4 . 5 ? Why or why not? 27. Suppose a very large pasture has a large number of cows, with an average rate of 1 cow per 100 m 2 . Would the number of cows in a randomly selected 100 m 2 area of this pasture follow a Poisson distribution? 8 The Geometric Distribution 28. Suppose a patient at a hospital in Canada is in dire need of a plasma transfusion. The person has type AB blood, and people with type AB blood can receive plasma only from other individuals with type AB blood. The hospital is out of AB plasma, but they have a long list of possible donors of unknown blood type. The donors are Canadian, and approximately 3% of the Canadian population no d given so False a iii ai F n ii.si pbinomC2.100 o.oD o.aappoisCa.D o.aia6 a a a 02 1 0 o.o

has type AB blood. Suppose these potential donors can be thought of as a random sample from the Canadian population. (a) If potential donors are tested for the AB blood type, what is the probability the first donor with type AB blood occurs on the fifth person tested? (b) If potential donors are tested for the AB blood type, what is the probability the first donor with type AB blood occurs on or before the third person tested? (c) If potential donors are tested for the AB blood type, what is the probability the first donor with type AB blood occurs after the 30th person tested? (d) What is the mean number of potential donors that must be tested in order to find one with type AB blood? (e) What is the standard deviation of the number of potential donors that must be tested in order to find one with type AB blood? 29. Tim Horton’s co ff ee shop chain has long held a “Roll up the Rim” contest, where purchasers of a hot beverage can roll up the rim of the cup, possibly revealing that the cup is a prize winner. The probability of winning a prize on each cup changes slightly from year to year, but it is often approximately 1 6 . Suppose that while the contest is running, you often buy cups of co ff ee from Tim Horton’s. (a) What is the probability your first winning cup comes on the fourth cup that you purchase? (While the probability of winning changes a tiny amount from cup to cup, as cups are revealed as winners or non-winners, there are millions of cups so the changes in probability are minuscule. For the purposes of these questions, assume that the cups can be considered independent.) (b) What is the probability your first win comes on or before the fourth cup? (c) What is the probability your first win comes after the 20th cup? (d) On average, how many cups will need to be purchased in order to get the first winner? 30. Which of the following statements are true? (a) The mean of the geometric distribution is 1 p . (b) The most likely value of a geometric random variable is always 1. (c) If p > 0 . 5 , then P ( X = 2) < P ( X = 3) . (d) The variance of the geometric distribution is always 1. (e) A geometric random variable can take on a countably infinite number of possible values. (f) If X has a geometric distribution, then Y = 1 X also has a geometric distribution. 9 The Negative Binomial Distribution 31. In his younger days, your author frequently practiced his golf game. While practicing around the greens, he would often play a game in which he had to hole a certain number of chips before allowing himself to leave. Suppose on a certain afternoon, he decided that he would allow himself to leave once he chipped-in 4 times. (He must hole out on a chip 4 times before leaving.) Suppose that his probability of chipping-in on any individual chip is approximately 0.05, and that the chips can be considered independent. (While the probability of chipping-in would vary a little from chip to chip, and the chips would not truly be independent, these assumptions provide a reasonable approximate model.) (a) What is the probability that the fourth chip-in occurs on the 20th attempt?

(b) In the next 20 spins of the wheel, what is the probability the ball lands in a red slot 14 times, a black slot 3 times, and the green slot 3 times? (c) In the next 20 spins of the wheel, what is the probability the ball lands in a red slot exactly 11 times? (d) What is the mean number of times the ball lands in a red slot in the next 20 spins? 35. Suppose X 1 , X 2 , . . . , X k have a multinomial distribution with parameters n and p 1 , p 2 , . . . , p k . Which of the following statements are true? (a) X 2 has a binomial distribution with parameters n and p 2 . (b) E ( X i ) = np i . (c) V ar ( X i ) = np i (1 - p i ) . (d) P k i =1 p i = 1 . 11 Chapter Exercises 11.1 Basic Calculations 36. Consider the following discrete probability distribution: x - 5 10 50 100 p ( x ) 1 c 1 c 8 c 10 c Where c is a constant. (a) What is the value of c ? (b) What is P ( X > 21) ? (c) What is P ( X > 21 | X < 60) ? 37. Consider the following discrete probability distribution for a random variable X . Note that the largest value of X and its corresponding probability are missing. x 10 20 30 ? p ( x ) 0 . 1 0 . 2 0 . 3 ? If E ( X ) = 60 , what is the missing value of X ? 38. The probability distribution for a random variable X is given by x 10 11 12 13 P ( X = x ) 0.4 0.3 0.2 ? (a) What is the value of the missing probability? (b) What is P ( X > 11) ? (c) What is the mean of this distribution? (d) What is the variance? 39. Consider again the discrete probability distribution given in Question 38 .

(a) Suppose 5 values are randomly sampled from this distribution. What is the probability that exactly three 10s are selected? (b) Suppose 5 values are randomly sampled from this distribution. What is the probability that at least one 10 is selected? (c) Values are sampled from this distribution repeatedly. What is the probability that the first 11 appears after the sixth sampled value? 40. Consider again the discrete probability distribution given in Question 38 . Suppose 20 values are randomly sampled from this distribution. (a) What is the probability that the number of 10s is less than 8? (b) What is the probability that the number of 10s is between 2 and 6 (inclusive)? (c) What is the expected number of 10s? 41. Let X represent the number of heads that come up when a fair coin is tossed 20 times. Let Y represent the number of fours that come up when a six-sided die is tossed 18 times. It is reasonable to assume that X and Y are independent. (a) What is the expected value of X + Y ? (b) What is the standard deviation of X - Y ? 42. Suppose a certain type of lottery ticket costs $1. Let the random variable X represent the payout on a single ticket. The distribution of X is given in the following table. x 0 1 10 100 p ( x ) 0.689 0.30 0.010 0.001 (a) Find the probability of making a net profit (including the cost of the ticket). (b) If two tickets are purchased, and the tickets can be assumed to be independent, what is the probability the two tickets have a combined payout of $100? (c) What is the expected value of the payout? (d) What is the standard deviation of the payout? (e) Given the payout is non-zero, what is the probability the payout is $100? (f) What is the lottery’s expected profit if 1000 of these tickets are sold? 11.2 Concepts 43. Show that E [( X - μ ) 2 ] = E ( X 2 ) - [ E ( X )] 2 . (You may use the property that E ( a + bX ) = a + bE ( X ) .) 44. Is the following statement true? Justify your response. Since E ( X 2 ) = E ( XX ) = E ( X ) E ( X ) , X is independent of itself. 45. Are the following statements true or false? (a) If X and Y are two random variables that both have a mean of 0, X and Y must be independent. (b) If E ( X ) > E ( Y ) , then V ar ( X ) > V ar ( Y ) . (c) If E ( X ) = 1 , then E ( X 2 ) = 1

(b) The mean of a Poisson random variable can be negative. (c) If X represents the number of students in a randomly selected 100 m 2 area of a university campus at 3:00 pm on a Tuesday afternoon, then X has a Poisson distribution. (d) If p > 0 , then the variance of a binomial random variable is less than its mean. (e) If X is a binomial random variable with n = 10 and p = 0 . 1 , and Y is a Poisson random variable with λ = 2 , then X - Y has a binomial distribution. 11.3 Applications 52. The insect Megamelus scutellaris can feed on the sap of the water hyacinth, and it has been used as a method of biocontrol for this invasive plant species. M. scutellaris mates on the water hyacinth, and females create oviposition scars on the plant, laying one or more eggs in each scar. The approximate distribution of the number of eggs per scar is given in the following table. (Loosely based on a study by Sosa et al. ( 2005 ).) x 1 2 3 4 p ( x ) 0 . 38 0 . 52 0 . 05 0 . 05 (a) What is the expected value of the number of eggs in a randomly selected oviposition scar? In other words, if we let X represent the number of eggs, what is E ( X ) ? (b) What is the standard deviation of the number of eggs? (c) What is the expected value of the natural log of the number of eggs? That is, what is E [ ln ( X )] ? (d) What is E ( X 2 ) ? 53. The Ontario Lottery Corporation runs a “Daily Keno” lottery in which 20 numbers are randomly selected without replacement from the integers 1 through 70. Before the drawing, keno players choose a set of numbers (anywhere from 2 to 10 numbers), and their ticket is a winner if enough of their numbers are randomly chosen. The following table represents the possible payouts and their probabilities of occurring for a $1 six-number ticket. 2 Number correct 4 or fewer 5 6 Payout 0 $25 $1000 Probability 0.9937921 0.0059123 0.0002956 (a) Verify the probabilities given in the table. (b) What is the expectation of the payout on a $1 six-number ticket? (c) What is the standard deviation of the payout on a $1 six-number ticket? (d) For every $1 spent on this type of ticket, on average how much is the government keeping? 54. Consider again the information in Question 53 . Suppose that you buy a single six-number keno ticket in each of the next 50 drawings. (a) What is the probability that exactly one ticket wins $25? (b) What is the probability that at least one ticket wins $25? (c) What is the probability that at least one ticket wins a prize? (d) What is the expected number of tickets that will win a prize? (e) What is the standard deviation of the number that will win a prize? 2 The listed payouts were extracted from http://www.olg.ca/lotteries/games/howtoplay.do?game=daily_keno#pt on January 23, 2015. The probabilities have been rounded to the 7th decimal place.

55. You have an agreement with a cheese supplier, which states that he will sell you 1,000 kg of mozzarella cheese for $7000. The supplier is a bit of a gambler and on March 10 o ff ers you an alternative arrangement: If it does not snow on April 10 you pay only $5,000, but if it snows that day you pay $100,000 instead. Suppose the probability of snowfall on April 10 is 0.02. (a) If you choose the alternative arrangement, how much will you owe the supplier on average? (b) Which deal is cheaper on average? (c) Give at least one reason why it is not always wise to choose the option that results in the greatest expected value. 56. A manufacturer needs to ship an expensive part across the country. The part is very fragile, and if it breaks it will cost the manufacturer $25,000. A company o ff ers insurance against breakage at a cost of $1,000. (a) If the probability of breakage is 0.01, what is the expected profit for the insurance company? (b) What is the standard deviation of the insurance company’s payout? (c) What probability of breakage makes this a break-even proposition for the manufacturer? 57. People in need of a bone marrow transplant need to find a willing donor that is closely matched in human leukocyte antigens (HLA). The genes that govern HLA are located on a single chromosome (chromosome 6). Since chromosomes are inherited from a person’s father and mother, any full sibling of a person has a 1 4 chance of being an HLA-identical match. Suppose that a person in need of a bone marrow transplant has 6 siblings (none of whom are the result of a multiple birth). (a) What is the distribution of the number of their siblings that are an HLA-identical match? (b) What is the probability that at least one of their siblings is an HLA-identical match? (c) What is the probability that at least two of their siblings are an HLA-identical match? (d) What is the mean number of siblings that are an HLA-identical match? 58. For some types of surveys conducted by random digit dialling, if the call is answered by an adult at a residence, there is approximately a 9% chance that the person will complete the survey. 3 Suppose you work at a call centre that conducts a survey of this type. For the following questions, “call” refers to a call that is answered by an adult at a residence. (a) In the next 100 calls, what is the probability that exactly 8 people complete the survey? (b) In the next 100 calls, what is the probability that 8, 9, or 10 people complete the survey? (c) In the next 100 calls, what is the probability that at least 2 people complete the survey? (d) What is the mean number of surveys that will be completed in the next 100 calls ? (e) What is the standard deviation of the number of surveys that will be completed in the next 100 calls ? 59. Consider again the information in Question 58 . Suppose you will get to leave for the day once you get the survey completed by 4 adults. For the following questions, “call” refers to a call that is answered by an adult at a residence. (a) What is the probability that the 4th completed survey occurs on the 50th call? 3 Estimated from information on the Public Works and Government Services Canada website http://www.tpsgc-pwgsc. gc.ca/rop-por/rapports-reports/telephone/introduction-eng.html . Accessed January 2015.

(a) What is the probability that there are no fatal crashes involving U.S. commercial airlines in the next year? (b) What is the probability that there are 2 or 3 fatal crashes involving U.S. commercial airlines in the next year? (c) What is the probability that there are more than 4 fatal crashes involving U.S. commercial airlines in the next year? (d) What is the standard deviation of the number of fatal crashes involving U.S. commercial airlines in the next year? 64. Consider again the information in Question 63 . (a) What is the probability that there is at least 1 fatal crash involving U.S. commercial airlines in the next 2 year period? (b) What is the probability that there is more than 1 fatal crash involving U.S. commercial airlines in the next 2 year period? (c) What is the mean number of fatal crashes involving U.S. commercial airlines in the next 2 year period? (d) What is the standard deviation of the number of fatal crashes involving U.S. commercial airlines in the next 2 year period? 65. According to Wikipedia, objects with a diameter in excess of 1 km hit the earth at a rate of approx- imately two every million years. Assume that the number of impacts in any given time period has (approximately) a Poisson distribution. (a) What is the approximate probability that there is at least one impact of this size in a given 100,000 year period? (b) What is the approximate probability that there is at least one impact of this size in a given 10,000 year period? 66. In a famous series of genetics experiments, Gregor Mendel investigated inheritance in sweet peas. In one experiment, he crossed one pure line of peas that had round yellow seeds with another pure line that had wrinkled green peas. A round shape and yellow colour are dominant traits, so the peas resulting from the first generation cross all had round yellow seeds (they had the round/yellow phenotype ). This first generation was then self-crossed. For the second generation, under Mendelian inheritance with independent assortment, a 9:3:3:1 ratio of phenotypes would be expected. In other words, the distribution of phenotypes would be: Phenotype Round/Yellow Round/Green Wrinkled/Yellow Wrinkled/Green Probability 9 16 3 16 3 16 1 16 Suppose we examine 50 pea plants from this type of cross. (a) What is the distribution of the number of round/yellow plants? (b) What is the mean number of round/yellow plants? (c) What is the standard deviation of the number of round/yellow plants? (d) What is the probability that there are exactly 4 wrinkled/green plants? (e) What is the probability that there are 28 round/yellow plants, 12 round/green plants, 6 wrin- kled/yellow plants, and 4 wrinkled/green plants?

11.4 Extra Practice Questions 67. Suppose that for a binomial random variable X , n = 10 and μ = 2 . 4 . (a) What is p ? (b) What is P ( X = 2) ? (c) What is σ ? 68. A certain coin sold by a novelty store has a probability of coming up heads of 2 3 . Let X be a random variable representing the total number of heads when this coin is tossed twice. (a) What is the mean of X ? (b) What is the standard deviation of X ? 69. According to the Pocket World in Figures by the Economist , 17.8% of the Canadian population is over 60 years old. Suppose 20 Canadians are randomly selected. (a) What is the distribution of the number of people in the sample that are over 60 years old? (b) What is the probability that at least 2 are over 60 years old? (c) What is the probability that exactly 2 are over 60 years old? (d) What is the probability that none are over 60 years old? (e) What is the expected number of people that are over 60 years old? 70. Approximately 60% of Canadian adults voted in the last federal election. Suppose we draw a random sample of five people who were Canadian adults at the time of the last election. (a) What is the distribution of the number of people in the sample that voted in the last federal election? (b) What is the probability that exactly 4 voted in the last federal election? (c) What is the probability that four or more voted in the last federal election? (d) Given at least four voted, what is the conditional probability that all 5 voted? (e) What is the mean of the number that voted in the last election? 71. A 2011 report by Statistics Canada showed the unemployment rate in Canada was approximately 7.6% (7.6% of the labour force was out of work). If 14 members of the Canadian labour force were randomly selected during this time period: (a) What is the probability exactly three were out of work? (b) What is the probability more than 2 were out of work? (c) Given at least 2 of the 14 were out of work, what is the probability at least 1 of the 14 was out of work? (d) Given at least 3 of the 14 were out of work, what is the probability no more than one of the 14 was out of work? 72. Approximately 35% of cars that are at least 13 years old fail the Ontario Drive Clean emissions test. Suppose we draw a random sample of 10 emissions tests on cars that are at least 13 years old. (a) What is the probability that exactly 2 fail? (b) What is the probability that no more than 2 fail? (c) What is the probability that more than 2 fail?

machine claims that for a certain type of weld, each weld has a 90% chance of being successful. Suppose that the company interested in the welder decides to test it by having it perform welds until it fails for the first time. Assume that the producer’s claim is true (each weld will be successful with probability 0.90 and unsuccessful with probability 0.10), and that the trials are independent. (a) What is the distribution of the number of welds needed to get the first unsuccessful weld? (b) What is the probability that the first unsuccessful weld occurs on the third trial? (c) What is the probability that the first unsuccessful weld occurs on or before the third trial? (d) What is the probability that the first unsuccessful weld occurs after the 13th trial? 79. Consider again the information in Question 78 . Suppose the company purchases the machine and uses it whenever a weld of the appropriate type is required. (Assume that the probability of a successful weld is always 0.90 and that the trials are independent.) (a) What is the probability that the third unsuccessful weld occurs on the 22nd welding job? (b) In the next 20 welds, what is the probability that exactly 3 are unsuccessful? (c) In the next 20 welds, what is the probability that at least 3 are unsuccessful? (d) What is the mean number of unsuccessful welds in a sample of 20 welds? 80. Suppose that you get a stimulating job inspecting shipments of light bulbs that arrive at a factory. Your job is to take out one randomly selected bulb and test it. If the bulb works, the shipment is accepted. If the bulb malfunctions then the shipment is rejected. Unknown to you, in reality the probability that any randomly selected bulb malfunctions is 0.04, and the shipments can be considered to be independent. (a) What is the probability that the first shipment you must reject occurs on the 10th shipment? (b) What is the probability that the first shipment you must reject occurs after the 10th shipment? (c) In the first 60 shipments, what is the probability you must send back no more than one shipment? 81. A room contains 30 women and 55 men, and 5 of these people are randomly selected without replacement, (a) What is the exact probability of getting exactly 4 men in the sample? (b) Use the binomial distribution to approximate the probability of getting 4 men in the sample. 82. Suppose the distribution of dandelions in a large meadow is approximately Poisson, with a mean of 4 dandelions per square metre. A 1 m 2 area in this meadow is randomly selected. (a) What is the probability there are exactly 2 dandelions in the area? (b) What is the probability there are more than 2 dandelions in the area? (c) Given there is at least one dandelion in the area, what is the probability there are exactly three dandelions? (d) What is the standard deviation of the number of dandelions in the randomly selected area? 83. Suppose you design a new type of dental implant. You feel that if the implant is placed in a randomly selected dental patient, the probability that it does not take properly is 0.05. As part of a test of the new implant, you intend to place your implant into patients until you get 3 unsuccessful implants. (Assume for this question that your probability assessment is correct, and also assume that the patients can be considered independent.) (a) What is the probability that the first unsuccessful implant occurs on the fourth trial?

(b) What is the probability that the first unsuccessful implant occurs after the 11th trial? (c) What is the probability that the third unsuccessful implant occurs on the 28th trial? 84. A dollar store bin contains 15 rubber footballs, 20 rubber soccer balls, and 8 rubber volleyballs. Suppose 6 of these balls are randomly selected without replacement. (a) What is the probability that exactly 2 footballs are selected? (b) What is the expectation of the number of footballs selected? (c) What is the probability that no more than 1 soccer ball is selected? (d) What is the probability 3 footballs, 2 soccer balls, and 1 volleyball are selected? 85. A room contains 5 biomedical engineers, 7 computer engineers, 5 mechanical engineers, and 3 water resource engineers. (a) If 6 people are randomly selected without replacement, what is the probability there are exactly 3 computer engineers selected in the group of 6? (b) If 6 people are randomly selected without replacement, what is the probability there are exactly 2 biomedical engineers, 2 computer engineers, 1 mechanical engineer, and 1 water resource engineer in the sample? 86. A rental car company has 20 cars available: 4 Hondas, 6 Fords, 3 Toyotas, and 7 Chrysler vehicles. If they randomly pick 5 of these cars for inspection (without replacement), what is the probability the sample consists of 1 Honda, 1 Ford, 1 Toyota, and 2 Chrysler vehicles? 87. Consider the following distribution of a categorical variable. Category A B C D Probability 0.4 0.3 0.2 0.1 Suppose 5 values are sampled independently from this distribution. What is the probability there will be three A’s, one B, and one C? References Fayrouz et al. (2012). Relation between fingerprints and di ff erent blood groups. Journal of Forensic and Legal Medicine , 19:18–21. Sosa et al. (2005). Life history of Megamelus scutellaris with description of immature stages (Hemiptera: Delphacidae). Annals of the Entomological Society of America , 98(1):66–72.