Ch. 6 Discrete Probability Distributions 6.1 Discrete Random Variables 1 Distinguish between discrete and continuous random variables. Classify the following random variable according to whether it is discrete or continuous. 1) the number of bottles of juice sold in a cafeteria during lunch A) discrete B) continuous 2) the heights of the bookcases in a school library A) continuous B) discrete 3) the cost of a road atlas A) discrete B) continuous 4) the pressure of water coming out of a fire hose A) continuous B) discrete 5) the temperature in degrees Celsius on January 1st in Fargo, North Dakota A) continuous B) discrete 6) the number of goals scored in a hockey game A) discrete B) continuous 7) the speed of a car on a New York tollway during rush hour traffic A) continuous B) discrete 8) the number of emails received on any given day A) discrete B) continuous 9) the age of the oldest dog in a kennel A) continuous B) discrete 10) the number of pills in an aspirin bottle A) discrete B) continuous Provide an appropriate response. 11) The peak shopping time at home improvement store is between 8:00am-11:00 am on Saturday mornings. Management at the home improvement store randomly selected 115 customers last Saturday morning and decided to observe their shopping habits. They recorded the number of items that each of the customers purchased as well as the total time the customers spent in the store. Identify the types of variables recorded by the home improvement store. A) number of items - discrete; total time - continuous B) number of items - continuous; total time - continuous C) number of items - continuous; total time - discrete D) number of items - discrete; total time - discrete 12) The number of violent crimes committed in a day possesses a distribution with a mean of 1.5 crimes per day and a standard deviation of four crimes per day. A random sample of 140 days was observed, and the sample mean number of crimes for the sample was calculated. The data that was collected in this experiment could be measured with a random variable. A) discrete B) continuous Page 1
13) A random variable is A) a numerical measure of the outcome of a probability experiment. B) generated by a random number table. C) the variable for which an algebraic equation is solved. D) a qualitative attribute of a population. 2 Identify discrete probability distributions. Provide an appropriate response. 14) Given the table of probabilities for the random variable x, does this form a probability distribution? Answer Yes or No. x 5 10 15 20 P(x) 0.10-0.30 0.50 0.70 A) No B) Yes 15) Given the table of probabilities for the random variable x, does this form a probability distribution? Answer Yes or No. x 0 1 2 3 4 P(x) 0.02 0.07 0.22 0.27 0.42 A) Yes B) No 16) Consider the discrete probability distribution to the right when answering the following question. Find the probability that x equals 4. x 3 4 6 8 P(x) 0.04? 0.07 0.31 A) 0.58 B) 0.42 C) 2.32 D) 1.68 17) Consider the discrete probability distribution to the right when answering the following question. Find the probability that x exceeds 5. x 3 5 7 8 P(x) 0.26? 0.02 0.28 A) 0.3 B) 0.74 C) 0.7 D) 0.44 18) An Apple Pie Company knows that the number of pies sold each day varies from day to day. The owner believes that on 50% of the days she sells 100 pies. On another 25% of the days she sells 150 pies, and she sells 200 pies on the remaining 25% of the days. To make sure she has enough product, the owner bakes 200 pies each day at a cost of $2 each. Assume any pies that go unsold are thrown out at the end of the day. If she sells the pies for $3 each, find the probability distribution for her daily profit. A) B) C) D) Profit P(profit) Profit P(profit) Profit P(profit) Profit P(profit) -$100 0.5 $50 0.25 $200 0.25 $300 0.5 $450 0.25 $600 0.25 $100 0.5 $250 0.25 $400 0.25 19) The sum of the probabilities of a discrete probability distribution must be A) equal to one. B) between zero and one. C) greater than one. D) less than or equal to zero. $100 0.5 $150 0.25 $200 0.25 Page 2
3 Graph discrete probability distributions. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 20) The random variable x represents the number of boys in a family of three children. Assuming that boys and girls are equally likely, (a) construct a probability distribution, and (b) graph the probability distribution. 21) The random variable x represents the number of tests that a pet entering an animal shelter will have along with the corresponding probabilities. Graph the probability distribution. x P(x) 0 3 17 1 5 17 2 6 17 3 2 17 4 1 17 22) The random variable x represents the number of credit cards that students have along with the corresponding probabilities. Graph the probability distribution. x P(x) 0 0.49 1 0.05 2 0.32 3 0.07 4 0.07 23) In an Italian cafe, the following probability distribution was obtained. The random variable x represents the number of toppings for a large pizza. Graph the probability distribution. x P(x) 0 0.30 1 0.40 2 0.20 3 0.06 4 0.04 24) Use the frequency distribution to (a) construct a probability distribution for the random variable x which represents the number of cars per family in a town of 1000 families, and (b) graph the probability distribution. Cars Families 0 125 1 428 2 256 3 108 4 83 Page 3
4 Compute and interpret the mean and standard deviation of a discrete random variable. Provide an appropriate response. 25) Calculate the mean for the discrete probability distribution shown here. x 2 5 9 10 P(x) 0.09 0.07 0.22 0.62 A) 8.71 B) 6.5 C) 26 D) 2.1775 26) A lab orders a shipment of 100 rats a week, 52 weeks a year, from a rat supplier for experiments that the lab conducts. Prices for each weekly shipment of rats follow the distribution below: Price $10.00 $12.50 $15.00 Probability 0.25 0.15 0.6 Suppose the mean cost of the rats turned out to be $13.38 per week. Interpret this value. A) The average cost for all weekly rat purchases is $13.38. B) Most of the weeks resulted in rat costs of $13.38. C) The median cost for the distribution of rat costs is $13.38. D) The rat cost that occurs more often than any other is $13.38. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 27) Calculate the mean for the discrete probability distribution shown here. x 2 6 10 15 P(x) 0.2 0.3 0.3 0.2 28) A baseball player is asked to swing at pitches in sets of four. The player swings at 100 sets of 4 pitches. The probability distribution for making a particular number of hits is given below. Determine the mean for this discrete probability distribution. x 0 1 2 3 4 P(x) 0.02 0.07 0.22 0.27 0.42 A) 3 B) 4 C) 2 D) 3.5 29) The produce manager at a farmerʹs market was interested in determining how many oranges a person buys when they buy oranges. He asked the cashiers over a weekend to count how many oranges a person bought when they bought oranges and record this number for analysis at a later time. The data is given below in the table. The random variable x represents the number of oranges purchased and P(x) represents the probability that a customer will buy x apples. Determine the mean number of oranges purchased by a customer. x 1 2 3 4 5 6 7 8 9 10 P(x) 0.05 0.19 0.20 0.25 0.12 0.10 0 0.08 0 0.01 A) 3.97 B) 5.50 C) 3 D) 4 Page 4
30) A random number generator is set to generate single digits between 0 and 9. One hundred and fifty random numbers are generated. The probability distribution for this random number generator is given below. What is the mean of this distribution? x 0 1 2 3 4 5 6 7 8 9 P(x) 0.09 0.12 0.11 0.08 0.09 0.13 0.10 0.07 0.10 0.11 A) 4.5 B) 6.6 C) 5 D) 7 31) A seed company has a test plot in which it is testing the germination of a hybrid seed. They plant 50 rows of 40 seeds per row. After a two-week period, the researchers count how many seed per row have sprouted. They noted that least number of seeds to germinate was 33 and some rows had all 40 germinate. The germination data is given below in the table. The random variable x represents the number of seed in a row that germinated and P(x) represents the probability of selecting a row with that number of seed germinating. Determine the mean number of seeds per row that germinated. x 33 34 35 36 37 38 39 40 P(x) 0.02 0.06 0.10 0.20 0.24 0.26 0.10 0.02 A) 36.9 B) 36.5 C) 36 D) 0.13 32) A manager asked her employees how many times they had given blood in the last year. The results of the survey are given below. The random variable x represents the number of times a person gave blood and P(x) represents the probability of selecting an employee who had given blood that percent of the time. What is the mean number of times a person gave blood based on this survey? x 0 1 2 3 4 5 6 P(x) 0.30 0.25 0.20 0.12 0.07 0.04 0.02 A) 1.6 B) 3.0 C) 2.0 D) 0.14 33) The random variable x represents the number of girls in a family of three children. Assuming that boys and girls are equally likely, find the mean and standard deviation for the random variable x. A) mean: 1.50; standard deviation: 0.87 B) mean: 2.25; standard deviation: 0.87 C) mean: 1.50; standard deviation: 0.76 D) mean: 2.25; standard deviation: 0.76 34) The random variable x represents the number of tests that a patient entering a clinic will have along with the corresponding probabilities. Find the mean and standard deviation for the random variable x. x P(x) 0 3 17 1 5 17 2 6 17 3 2 17 4 1 17 A) mean: 1.59; standard deviation: 1.09 B) mean: 1.59; standard deviation: 3.72 C) mean: 2.52; standard deviation: 1.93 D) mean: 3.72; standard deviation: 2.52 Page 5
35) The random variable x represents the number of computers that families have along with the corresponding probabilities. Find the mean and standard deviation for the random variable x. x P(x) 0 0.49 1 0.05 2 0.32 3 0.07 4 0.07 A) mean: 1.18; standard deviation: 1.30 B) mean: 1.39; standard deviation: 0.64 C) mean: 1.39; standard deviation: 0.80 D) mean: 1.18; standard deviation: 0.64 36) In a sandwich shop, the following probability distribution was obtained. The random variable x represents the number of condiments used for a hamburger. Find the mean and standard deviation for the random variable x. x P(x) 0 0.30 1 0.40 2 0.20 3 0.06 4 0.04 A) mean: 1.14; standard deviation: 1.04 B) mean: 1.54; standard deviation: 1.30 C) mean: 1.30; standard deviation: 2.38 D) mean: 1.30; standard deviation: 1.54 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 37) From the probability distribution, find the mean and standard deviation for the random variable x, which represents the number of bicycles per household in a town of 1000 households. x P(x) 0 0.125 1 0.428 2 0.256 3 0.108 4 0.083 38) A baseball player is asked to swing at pitches in sets of four. The player swings at 100 sets of 4 pitches. The probability distribution for hitting a particular number of pitches is given below. Determine the standard deviation for this discrete probability distribution. x 0 1 2 3 4 P(x) 0.02 0.07 0.22 0.27 0.42 A) 1.05 B) 1.10 C) 1.21 D) 0.28 39) The owner of a farmerʹs market was interested in determining how many oranges a person buys when they buy oranges. He asked the cashiers over a weekend to count how many oranges a person bought when they bought oranges and record this number for analysis at a later time. The data is given below in the table. The random variable x represents the number of oranges purchased and P(x) represents the probability that a customer will buy x oranges. Determine the variance of the number of oranges purchased by a customer. x 1 2 3 4 5 6 7 8 9 10 P(x) 0.05 0.19 0.20 0.25 0.12 0.10 0 0.08 0 0.01 A) 3.57 B) 1.95 C) 3.97 D) 0.56 Page 6
40) A manager at a local company asked his employees how many times they had given blood in the last year. The results of the survey are given below. The random variable x represents the number of times a person gave blood and P(x) represents the probability of selecting an employee who had given blood that percent of the time. What is the standard deviation for the number of times a person gave blood based on this survey? x 0 1 2 3 4 5 6 P(x) 0.30 0.25 0.20 0.12 0.07 0.04 0.02 A) 1.54 B) 1.82 C) 1.16 D) 2.23 41) A seed company has a test plot in which it is testing the germination of a hybrid seed. They plant 50 rows of 40 seeds per row. After a two-week period, the researchers count how many seed per row have sprouted. They noted that least number of seeds to germinate was 33 and some rows had all 40 germinate. The germination data is given below in the table. The random variable x represents the number of seed in a row that germinated and P(x) represents the probability of selecting a row with that number of seed germinating. Determine the standard deviation of the number of seeds per row that germinated. x 33 34 35 36 37 38 39 40 P(x) 0.02 0.06 0.10 0.20 0.24 0.26 0.10 0.02 A) 1.51 B) 7.13 C) 36.86 D) 6.07 5 Interpret the mean of a discrete random variable as an expected value. Provide an appropriate response. 42) A lab orders a shipment of 100 rats a week, 52 weeks a year, from a rat supplier for experiments that the lab conducts. Prices for each weekly shipment of rats follow the distribution below: Price $10.00 $12.50 $15.00 Probability 0.3 0.25 0.45 How much should the lab budget for next yearʹs rat orders assuming this distribution does not change. (Hint: find the expected price.) A) $669.50 B) $12.88 C) $1288.00 D) $3,481,400.00 43) Mamma Temte bakes six pies a day that cost $2 each to produce. On 38% of the days she sells only two pies. On 22% of the days, she sells 4 pies, and on the remaining 40% of the days, she sells all six pies. If Mama Temte sells her pies for $4 each, what is her expected profit for a dayʹs worth of pies? [Assume that any leftover pies are given away.] A) $4.16 B) $16.16 C) -$8.00 D) -$7.96 44) A local bakery has determined a probability distribution for the number of cheesecakes that they sell in a given day. The distribution is as follows: Number sold in a day 0 5 10 15 20 Prob (Number sold) 0.22 0.24 0.13 0.25 0.16 Find the number of cheesecakes that this local bakery expects to sell in a day. A) 9.45 B) 9.67 C) 10.55 D) 10 Page 7
45) A dice game involves throwing three dice and betting on one of the six numbers that are on the dice. The game costs $5 to play, and you win if the number you bet appears on any of the dice. The distribution for the outcomes of the game (including the profit) is shown below: Number of dice with your number Profit Probability of Observing 0 -$5 125/216 1 $5 75/216 2 $7 15/216 3 $15 1/216 Find your expected profit from playing this game. A) -$0.62 B) $0.50 C) $5.17 D) $2.88 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 46) A legendary football coach was known for his winning seasons. He consistently won nine or more games per season. Suppose x equals the number of games won up to the halfway mark (six games) in a 12-game season. If this coach and his team had a probability p = 0.6 of winning any one game (and the winning or losing of one game was independent of another), then the probability distribution of the number x of winning games in a series of six games is: x P(x) 0 0.004096 1 0.036864 2 0.138240 3 0.276480 4 0.311040 5 0.186624 6 0.046656 Find the expected number of winning games in the first half of the season for this coachʹs football teams. 47) On one busy holiday weekend, a national airline has many requests for standby flights at half of the usual one-way air fare. However, past experience has shown that these passengers have only about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as a standby, their only other choice is to fly first class on the next flight out. Suppose that the usual one-way air fare to a certain city is $100 and the cost of flying first class is $490. Should a passenger who wishes to fly to this city opt to fly as a standby? [Hint: Find the expected cost of the trip for a person flying standby.] 48) An automobile insurance company estimates the following loss probabilities for the next year on a $25,000 sports car: Total loss: 0.001 50% loss: 0.01 25% loss: 0.05 10% loss: 0.10 Assuming the company will sell only a $500 deductible policy for this model (i.e., the owner covers the first $500 damage), how much annual premium should the company charge in order to average $500 profit per policy sold? 49) True or False: The expected value of a discrete probability distribution may be negative. A) True B) False Page 8
50) In a carnival game, a person wagers $2 on the roll of two dice. If the total of the two dice is 2, 3, 4, 5, or 6 then the person gets $4 (the $2 wager and $2 winnings). If the total of the two dice is 8, 9, 10, 11, or 12 then the person gets nothing (loses $2). If the total of the two dice is 7, the person gets $0.75 back (loses $0.25). What is the expected value of playing the game once? A) -$0.04 B) -$0.42 C) $2.00 D) $0.00 51) In the American version of the Game Roulette, a wheel has 18 black slots, 8 red slots and 2 green slots. All slots are the same size. In a carnival game, a person wagers $2 on the roll of two dice. A person can wager on either red or black. Green is reserved for the house. If a player wagers $5 on either red or black and that color comes up, they win $10 otherwise they lose their wager. What is the expected value of playing the game once? A) -$0.26 B) -$0.50 C) $0.26 D) $0.50 6.2 The Binomial Probability Distribution 1 Determine whether a probability experiment is a binomial experiment. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Decide whether the experiment is a binomial experiment. If it is not, explain why. You observe the gender of the next 150 babies born at a local hospital. The random variable represents the number of boys. 2) Decide whether the experiment is a binomial experiment. If it is not, explain why. You draw a marble 1000 times from a bag with three colors of marbles. The random variable represents the color of marble that is drawn. 3) Decide whether the experiment is a binomial experiment. If it is not, explain why. In a game you spin a wheel that has 15 different letters 50 times. The random variable represents the selected letter on each spin of the wheel. 4) Decide whether the experiment is a binomial experiment. If it is not, explain why. Testing a cough suppressant using 340 people to determine if it is effective. The random variable represents the number of people who find the cough suppressant to be effective. 5) Decide whether the experiment is a binomial experiment. If it is not, explain why. Survey 650 investors to see how many different stocks they own. The random variable represents the number of different stocks owned by each investor. 6) Decide whether the experiment is a binomial experiment. If it is not, explain why. Survey 800 college students see whether they are enrolled as a new student. The random variable represents the number of students enrolled as new students. 7) Decide whether the experiment is a binomial experiment. If it is not, explain why. Each week, a man attends a club meeting in which he has a 34% chance of meeting a new member. The random variable is the number of times he meets a new member in 24 weeks. 8) Decide whether the experiment is a binomial experiment. If it is not, explain why. You test four flu medicines. The random variable represents the flue medicine that is most effective. 9) Decide whether the experiment is a binomial experiment. If it is not, explain why. Each week, a gambler plays blackjack at the local casino. The random variable is the number of times per week the player wins. 10) Decide whether the experiment is a binomial experiment. If it is not, explain why. Selecting five cards, one at a time without replacement, from a standard deck of cards. The random variable is the number of picture cards obtained. Page 9
11) True or False: The trials of a binomial experiment must be mutually exclusive of each other. A) False B) True 12) Which of the below is not a requirement for binomial experiment? A) The trials are mutually exclusive. B) The experiment is performed a fixed number of times. C) For each trial there are two mutually exclusive outcomes. D) The probability of success is fixed for each trial of the experiment. 13) If p is the probability of success of a binomial experiment, then the probability of failure is A) 1 - p B) -p C) x n D) n x 2 Compute probabilities of binomial experiments. Provide an appropriate response. 14) Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of exactly eight girls in ten births. A) 0.044 B) 0.8 C) 0.176 D) 0.08 15) In a recent survey, 60% of the community favored building a health center in their neighborhood. If 14 citizens are chosen, find the probability that exactly 5 of them favor the building of the health center. A) 0.041 B) 0.207 C) 0.357 D) 0.600 16) The probability that an individual has 20-20 vision is 0.13. In a class of 72 students, what is the probability of finding five people with 20-20 vision? A) 0.046 B) 0.069 C) 0.000 D) 0.13 17) According to insurance records a car with a certain protection system will be recovered 91% of the time. Find the probability that 5 of 6 stolen cars will be recovered. A) 0.337 B) 0.833 C) 0.91 D) 0.09 18) The probability that a football game will go into overtime is 18%. What is the probability that two of three football games will go to into overtime? A) 0.080 B) 0.18 C) 0.363 D) 0.0324 19) Fifty percent of the people that use the Internet order something online. Find the probability that only two of 10 Internet users will order something online. A) 0.044 B) 0.200 C) 0.001 D) 11.250 20) The probability that a house in an urban area will develop a leak is 6%. If 87 houses are randomly selected, what is the probability that none of the houses will develop a leak? A) 0.005 B) 0.060 C) 0.000 D) 0.001 21) Sixty-five percent of men consider themselves knowledgeable soccer fans. If 13 men are randomly selected, find the probability that exactly five of them will consider themselves knowledgeable fans. A) 0.034 B) 0.215 C) 0.65 D) 0.385 22) Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of at most three girls in ten births. A) 0.172 B) 0.300 C) 0.003 D) 0.333 Page 10
23) A quiz consists of 10 true or false questions. To pass the quiz a student must answer at least eight questions correctly. If the student guesses on each question, what is the probability that the student will pass the quiz? A) 0.055 B) 0.8 C) 0.20 D) 0.08 24) A quiz consists of 10 multiple choice questions, each with five possible answers, one of which is correct. To pass the quiz a student must get 60% or better on the quiz. If a student randomly guesses, what is the probability that the student will pass the quiz? A) 0.006 B) 0.060 C) 0.377 D) 0.205 25) A recent survey found that 70% of all adults over 50 wear sunglasses for driving. In a random sample of 10 adults over 50, what is the probability that at least six wear sunglasses? A) 0.850 B) 0.700 C) 0.200 D) 0.006 26) According to government data, the probability that an adult was never in a museum is 15%. In a random survey of 10 adults, what is the probability that two or fewer were never in a museum? A) 0.820 B) 0.002 C) 0.800 D) 0.200 27) According to government data, the probability that an adult was never in a museum is 15%. In a random survey of 10 adults, what is the probability that at least eight were in a museum? A) 0.820 B) 0.200 C) 0.002 D) 0.800 28) According to the Federal Communications Commission, 70% of all U.S. households have vcrs. In a random sample of 15 households, what is the probability that fewer than 13 have vcrs? A) 0.8732 B) 0.7 C) 0.1268 D) 0.5 29) According to the Federal Communications Commission, 70% of all U.S. households have vcrs. In a random sample of 15 households, what is the probability that the number of households with vcrs is between 10 and 12, inclusive? A) 0.5947 B) 0.4053 C) 0.7 D) 0.2061 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 30) A motel has a policy of booking as many as 150 guests in a building that holds 140. Past studies indicate that only 85% of booked guests show up for their room. Find the probability that if the motel books 150 guests, not enough seats will be available. 31) We believe that 80% of the population of all Calculus I students consider calculus an exciting subject. Suppose we randomly and independently selected 24 students from the population. If the true percentage is really 80%, find the probability of observing 23 or more of the students who consider calculus to be an exciting subject in our sample of 24. A) 0.033057 B) 0.004722 C) 0.028334 D) 0.966943 32) A psychic network received telephone calls last year from over 1.5 million people. A recent article attempts to shed some light onto the credibility of the psychic network. One of the psychic networkʹs psychics agreed to take part in the following experiment. Five different cards are shuffled, and one is chosen at random. The psychic will then try to identify which card was drawn without seeing it. Assume that the experiment was repeated 55 times and that the results of any two experiments are independent of one another. If we assume that the psychic is a fake (i.e., they are merely guessing at the cards and have no psychic powers), find the probability that they guess at least three correctly. A) 0.999497 B) 0.000494 C) 0.001917 D) 0.000434 Page 11
33) A history professor decides to give a 10-question true-false quiz. She wants to choose the passing grade such that the probability of passing a student who guesses on every question is less than 0.10. What score should be set as the lowest passing grade? A) 8 B) 6 C) 9 D) 7 34) A recent article in the paper claims that government ethics are at an all-time low. Reporting on a recent sample, the paper claims that 36% of all constituents believe their representative possesses low ethical standards. Assume that responses were randomly and independently collected. A representative of a district with 1,000 people does not believe the paperʹs claim applies to her. If the claim is true, how many of the representativeʹs constituents believe the representative possesses low ethical standards? A) 360 B) 36 C) 640 D) 964 35) A recent article in the paper claims that government ethics are at an all-time low. Reporting on a recent sample, the paper claims that 41% of all constituents believe their representative possesses low ethical standards. Suppose 20 of a representativeʹs constituents are randomly and independently sampled. Assuming the paperʹs claim is correct, find the probability that more than eight but fewer than 12 of the 20 constituents sampled believe their representative possesses low ethical standards. A) 0.372646 B) 0.245893 C) 0.593087 D) 0.596302 3 Compute the mean and standard deviation of a binomial random variable. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 36) Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Suppose that 950 couples each have a baby; find the mean and standard deviation for the number of boys in the 950 babies. 37) A quiz consists of 570 true or false questions. If the student guesses on each question, what is the mean number of correct answers? A) 285 B) 0 C) 570 D) 114 38) A quiz consists of 100 true or false questions. If the student guesses on each question, what is the standard deviation of the number of correct answers? A) 5 B) 0 C) 2 D) 7.07106781 39) A quiz consists of 20 multiple choice questions, each with five possible answers, only one of which is correct. If a student guesses on each question, what is the mean and standard deviation of the number of correct answers? A) mean: 4; standard deviation: 1.78885438 B) mean: 4; standard deviation: 2 C) mean: 10; standard deviation: 1.78885438 D) mean: 10; standard deviation: 3.16227766 40) The probability that an individual has 20-20 vision is 0.12. In a class of 70 students, what is the mean and standard deviation of the number with 20-20 vision in the class? A) mean: 8.4; standard deviation: 2.71882327 B) mean: 70; standard deviation: 2.71882327 C) mean: 8.4; standard deviation: 2.89827535 D) mean: 70; standard deviation: 2.89827535 41) A recent survey found that 79% of all adults over 50 wear sunglasses for driving. In a random sample of 80 adults over 50, what is the mean and standard deviation of those that wear sunglasses? A) mean: 63.2; standard deviation: 3.64307562 B) mean: 63.2; standard deviation: 7.94984277 C) mean: 16.8; standard deviation: 3.64307562 D) mean: 16.8; standard deviation: 7.94984277 Page 12
42) According to government data, the probability that an adult was never in a museum is 10%. In a random survey of 20 adults, what is the mean and standard deviation of the number that were never in a museum? A) mean: 2; standard deviation: 1.34164079 B) mean: 18 standard deviation: 1.41421356 C) mean: 18; standard deviation: 1.34164079 D) mean: 2; standard deviation: 1.41421356 43) According to insurance records, a car with a certain protection system will be recovered 85% of the time. If 500 stolen cars are randomly selected, what is the mean and standard deviation of the number of cars recovered after being stolen? A) mean: 425; standard deviation: 7.98435971 B) mean: 425; standard deviation: 63.75 C) mean: -2075: standard deviation: 7.98435971 D) mean: -2075: standard deviation: 63.75 44) The probability that a football game will go into overtime is 16%. In 120 randomly selected football games, what is the mean and the standard deviation of the number that went into overtime? A) mean: 19.2; standard deviation: 4.01596813 B) mean: 19.2; standard deviation: 4.38178046 C) mean: 18; standard deviation: 4.01596813 D) mean: 18; standard deviation: 4.38178046 45) In a recent survey, 80% of the community favored building a health center in their neighborhood. If 15 citizens are chosen, what is the mean number favoring the health center? A) 12 B) 15 C) 8 D) 10 46) In a recent survey, 80% of the community favored building a health center in their neighborhood. If 15 citizens are chosen, what is the standard deviation of the number favoring the health center? A) 1.55 B) 2.40 C) 0.98 D) 0.55 47) The probability that a house in an urban area will develop a leak is 5%. If 20 houses are randomly selected, what is the mean of the number of houses that developed leaks? A) 1 B) 2 C) 0.5 D) 1.5 48) A psychic network received telephone calls last year from over 1.5 million people. A recent article attempts to shed some light onto the credibility of the psychic network. One of the psychic networkʹs psychics agreed to take part in the following experiment. Five different cards are shuffled, and one is chosen at random. The psychic will then try to identify which card was drawn without seeing it. Assume that the experiment was repeated 35 times and that the results of any two experiments are independent of one another. If we assume that the psychic is a fake (i.e., they are merely guessing at the cards and have no psychic powers), how many of the 35 cards do we expect the psychic to guess correctly? A) 7 B) 6 C) 0 D) 5 4 Graph a binomial probability distribution. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 49) Draw the probability graph and label the mean for n = 6 and p = 0.4 50) Draw the probability graph and label the mean for n = 7 and p = 0.5 51) Draw the probability graph and label the mean for n = 8 and p = 0.3 52) Draw the probability graph and label the mean for n = 9 and p = 0.7 Page 13
6.3 The Poisson Probability Distribution 1 Determine if a probability experiment follows a Poisson process. Provide an appropriate response. 1) Given that a random variable x, the number of successes, follows a Poisson process, the probability of 2 or more successes in any sufficiently small subinterval is A) zero. B) one. C) any number between zero and one. D) none of these 2) Given that a random variable x, the number of successes, follows a Poisson process, then the probability of success for any two intervals of the same size A) is the same. B) are complementary. C) are reciprocals. D) none of these 3) Given that a random variable x, the number of successes, follows a Poisson process, then the number of successes in any interval is independent of the number of successes in any other interval provided the intervals A) are disjoint. B) overlap. C) have at least one element in common. D) are the same size and are independent. 2 Compute probabilities of a Poisson random variable. Provide an appropriate response. 4) A history professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of three students arrive. Use the Poisson distribution to find the probability that in a randomly selected office hour in the 10:30 a.m. time slot exactly four students will arrive. A) 0.1680 B) 0.0618 C) 0.1328 D) 0.0489 5) A help desk receives an average of four calls per hour on its toll-free number. For any given hour, find the probability that it will receive exactly three calls. Use the Poisson distribution. A) 0.1954 B) 0.5312 C) 271.0704 D) 0.2474 6) The local police department receives an average of two calls per hour. Use the Poisson distribution to determine the probability that in a randomly selected hour the number of calls is six. A) 0.0120 B) 0.0002 C) 0.0068 D) 0.0001 7) A dictionary contains 500 pages. If there are 200 typing errors randomly distributed throughout the book, use the Poisson distribution to determine the probability that a page contains exactly three errors. A) 0.0072 B) 0.0005 C) 0.1734 D) 0.0129 8) A customer service firm receives an average of three calls per hour on its toll-free number. For any given hour, find the probability that it will receive at least three calls. Use the Poisson distribution. A) 0.5768 B) 0.1891 C) 0.4232 D) 0.6138 9) An online retailer receives an average of five orders per 500 hits on its website. If it gets 100 hits on its website, find the probability of receiving at least two orders. Use the Poisson distribution. A) 0.2642 B) 0.1839 C) 0.9048 D) 0.9596 Page 14
10) A local animal rescue organization receives an average of 0.55 rescue calls per hour. Use the Poisson distribution to find the probability that during a randomly selected hour, the organization will receive fewer than two calls. A) 0.894 B) 0.106 C) 0.317 D) 0.087 11) The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 3. Find the probability that exactly four road construction projects are currently taking place in this city. A) 0.168031 B) 0.247261 C) 0.132766 D) 0.195367 12) The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 5. Find the probability that more than four road construction projects are currently taking place in the city. A) 0.559507 B) 0.734974 C) 0.440493 D) 0.265026 13) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 6.4. Find the probability that less than three accidents will occur next month on this stretch of road. A) 0.046324 B) 0.118919 C) 0.953676 D) 0.881081 14) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.5. Find the probability of observing exactly four accidents on this stretch of road next month. A) 0.044255 B) 3.983699 C) 1.111209 D) 100.027813 15) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 6.3. Find the probability that the next two months will both result in five accidents each occurring on this stretch of road. A) 0.023064 B) 0.151868 C) 0.303736 D) 0.008924 16) Suppose the number of babies born during an 8-hour shift at a hospitalʹs maternity wing follows a Poisson distribution with a mean of 4 an hour. Find the probability that seven babies are born during a particular 1-hour period in this maternity wing. A) 0.059540 B) 0.000001 C) 0.007443 D) 0.000012 17) Suppose the number of babies born during an 8-hour shift at a hospitalʹs maternity wing follows a Poisson distribution with a mean of 4 an hour. Some people believe that the presence of a full moon increases the number of births that take place. Suppose during the presence of a full moon, County Hospital experienced eight consecutive hours with more than five births. Based on this fact, comment on the belief that the full moon increases the number of births. A) The belief is supported as the probability of observing this many births would be 0.00000457. B) The belief is not supported as the probability of observing this many births is 0.215. C) The belief is supported as the probability of observing this many births would be 0.215. D) The belief is not supported as the probability of observing this many births is 0.00000457. 18) The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 8.5 tickets per day. Find the probability that less than six tickets are written on a randomly selected day from this distribution. A) 0.149597 B) 0.256178 C) 0.850403 D) 0.743822 Page 15
19) The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 9.3 tickets per day. Find the probability that exactly four tickets are written on a randomly selected day from this distribution. A) 0.028496 B) 1.513589 C) 5.708771 D) 303.227628 20) The number of goals scored at State College hockey games follows a Poisson distribution with a mean of 3 goals per game. Find the probability that a randomly selected State College hockey game would have more than three goals scored. A) 0.352768 B) 0.647232 C) 0.576810 D) 0.423190 21) The number of goals scored at State College hockey games follows a Poisson distribution with a mean of 5 goals per game. Find the probability that each of three randomly selected State College hockey games resulted in six goals being scored. A) 0.00312641 B) 0.43866845 C) 0.00000765 D) 0.56133155 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 22) A small life insurance company has determined that on the average it receives 5 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day. 23) The number of traffic accidents that occurs on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.9. Find the probability that less than two accidents will occur on this stretch of road during a randomly selected month. 24) Suppose the number of babies born during an eight-hour shift at a hospitalʹs maternity wing follows a Poisson distribution with a mean of 3 an hour. Find the probability that exactly seven babies are born during a randomly selected hour. 3 Find the mean and standard deviation of a Poisson random variable. Provide an appropriate response. 25) The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 4.5 tickets per day. Interpret the value of the mean. A) If we sampled all days, the arithmetic average number of tickets written would be 4.5 tickets per day. B) The number of tickets that is written most often is 4.5 tickets per day. C) Half of the days have less than 4.5 tickets written and half of the days have more than 4.5 tickets written. D) The mean has no interpretation since 0.5 ticket can never be written. 26) Suppose x is a random variable for which a Poisson probability distribution with λ = 6.9 provides a good characterization. Find μ for x. A) 6.9 B) 2.63 C) 3.5 D) 47.61 27) Suppose x is a random variable for which a Poisson probability distribution with λ = 10 provides a good characterization. Find σ for x. A) 3.16 B) 10 C) 5 D) 100 28) The number of goals scored at State College soccer games follows a Poisson process with a goal scored approximately every 18 minutes (a soccer game consists of 2 45 -minute halves). What is the mean number of goals scored during a game A) 5 B) 2.5 C) 0.11 D) 1.25 Page 16
29) The number of goals scored at State College soccer games follows a Poisson process with a goal scored approximately every 18 minutes (a soccer game consists of 2 45 -minute halves). What is the standard deviation of the mean number of goals scored during a game? A) 2.34 B) 1.58 C) 1.12 D) 1.27 30) The university police department keeps track of the number of tickets it write in a year. Last year the campus police wrote 1460 tickets. Ticket writing on campus follows a Poisson process. What is the mean number of tickets written per day by the campus police? A) 4 B) 2 C) 6.08 D) 5 31) The university police department keeps track of the number of tickets it write in a year. Last year the campus police wrote 1460 tickets. Ticket writing on campus follows a Poisson process. What is the standard deviation of the number of tickets written per day by the campus police? A) 2 B) 1.4 C) 2.47 D) 1.73 6.4 The Hypergeometric Probability Distribution (Online) 1 Determine whether a probability experiment is a hypergeometric experiment. Determine whether the probability experiment represents a hypergeometric probability experiment. If it does, determine the values of N, n, and k and list the possible values of the random variable X. 1) A jury is to be selected from a pool of 39 potential jurors. The defendant faces the death penalty if convicted. Of the potential jurors, 7 are opposed to the death penalty. The jury consists of 12 randomly selected jurors. The random variable X represents the number of jurors who oppose the death penalty. A) hypergeometric; N = 39, n = 12, k = 7, x = 0, 1, 2,..., 7 B) hypergeometric; N = 39, n = 7, k = 12, x = 0, 1, 2,..., 7 C) hypergeometric; N = 39, n = 12, k = 7, x = 0, 1, 2,...,12 D) not hypergeometric 2) An electronics store receives a shipment of 49 flat screen TVs of which 5 are defective. During the quality control inspection, 4 TVs are selected at random from the shipment for testing. The random variable X represents the number of defective computers in the sample A) hypergeometric; N = 49, n = 4, k = 5, x = 0, 1, 2, 3, 4 B) hypergeometric; N = 49, n = 5, k = 4, x = 0, 1, 2, 3, 4 C) hypergeometric; N = 49, n = 4, k = 5, x = 0, 1, 2,..., 5 D) not hypergeometric 3) A university must choose a team of 5 students to participate in a TV quiz show. The students will be chosen at random from a pool of 42 potential participants of whom 23 are women. The random variable X represents the number of women on the team. A) hypergeometric; N = 42, n = 5, k = 23, x = 0, 1, 2, 3, 4, 5 B) hypergeometric; N = 42, n = 23, k = 5, x = 0, 1, 2, 3, 4, 5 C) hypergeometric; N = 42, n = 5, k = 23, x = 0, 1, 2,..., 23 D) not hypergeometric 4) Of the 3200 students at a college, 280 are mature students. 50 students are selected at random from the 3200 students at the college and asked to participate in a survey. The questions in the survey relate to the studentʹs need for financial assistance. The random variable X represents the number of mature students in the sample. A) hypergeometric; N = 3200, n = 50, k = 280, x = 0, 1, 2,..., 50 B) hypergeometric; N = 3200, n = 280, k = 50, x = 0, 1, 2,..., 50 C) hypergeometric; N = 3200, n = 50, k = 280, x = 0, 1, 2,..., 280 D) not hypergeometric Page 17
2 Compute the probabilities of hypergeometric experiments. Provide an appropriate response. 5) A hypergeometric probability experiment is conducted with the given parameters. Compute the probability of obtaining x successes. N = 130, n = 10, k = 25, x = 3 A) 0.1965 B) 0.1914 C) 0.1867 D) 0.1722 6) A manufacturer receives a shipment of 190 laptop computers of which 10 are defective. To test the shipment, the quality control engineer randomly selects 15 computers from the shipment and tests them. The random variable X represents the number of defective computers in the sample. What is the probability of obtaining 2 defective computers? A) 0.1467 B) 0.1440 C) 0.1368 D) 0.1540 7) A university must choose a team of 6 students to participate in a TV quiz show. The students will be chosen from a pool of 32 potential participants of whom 14 are women. If the 6 students are chosen at random, what is the probability that the team will contain 2 women? A) 0.3073 B) 0.2766 C) 0.2874 D) 0.2587 Solve the problem. 8) In a lottery, a player selects six numbers between 1 and 37 inclusive. The six winning numbers (all different) are selected at random from the numbers 1-37. To win a prize, the player must match three or more of the winning numbers. What is the probability that the player matches exactly 3 numbers? A) 0.0387 B) 0.0322 C) 0.0451 D) 0.0483 9) An electronics store receives a shipment of 50 flat screen TVs of which 6 are defective. During the quality control inspection, 3 TVs are selected at random from the shipment for testing. The shipment will only be accepted if all 3 TVs pass the inspection. What is the probability that the shipment will be accepted? A) 0.6757 B) 0.6815 C) 0.0017 D) 0.001 10) A IRS auditor randomly selects 3 tax returns from 46 returns of which 10 contain errors. What is the probability that none of the returns she selects contains an error? A) 0.4704 B) 0.4793 C) 0.0103 D) 0.0079 11) Among the contestants in a competition are 45 women and 21 men. If 5 winners are randomly selected, what is the probability that they are all men? A) 0.00228 B) 0.01666 C) 0.14735 D) 0.02213 Provide an appropriate response. 12) A jury is to be selected from a pool of 38 potential jurors. The defendant faces the death penalty if convicted. Of the potential jurors, 6 are opposed to the death penalty and would not convict regardless of the evidence. The prosecutor knows that if even one juror opposes the death penalty, they will have no chance of getting a conviction. If none of the jurors opposes the death penalty they will have a chance of getting a conviction. What is the probability that none of the jurors opposes the death penalty, if the jury consists of 12 randomly selected jurors? A) 0.0834 B) 0.0876 C) 0.1272 D) 0.1208 Page 18
3 Compute the mean and standard deviation of a hypergeometric random variable. Compute the mean and standard deviation of the hypergeometric random variable X. 13) N = 100, n = 10, k = 35 A) μx = 3.5, σx = 1.44 B) μx = 3.5, σx = 1.51 C) μx = 0.35, σx = 1.44 D) μx = 4.2, σx = 1.51 14) In a lottery, a player must choose 6 numbers between 1 and 48 inclusive. Six balls are then randomly selected from an urn containing 48 balls numbered from 1 to 48. The random variable X represents the number of matching numbers. What are the mean and standard deviation of the random variable X? A) μx = 0.750, σx = 35.992 B) μx = 0.750, σx = 0.810 C) μx = 0.125, σx = 35.992 D) μx = 0.125, σx = 0.810 15) A manufacturer receives a shipment of 160 laptop computers of which 8 are defective. To test the shipment, the quality control engineer randomly selects 15 computers from the shipment and tests them. The random variable X represents the number of defective computers in the sample. What are the mean and standard deviation of the random variable X? A) μx = 0.750, σx = 0.806 B) μx = 0.750, σx = 0.844 C) μx = 0.050, σx = 0.806 D) μx = 0.750, σx = 0.650 Page 19
Ch. 6 Discrete Probability Distributions Answer Key 6.1 Discrete Random Variables 1 Distinguish between discrete and continuous random variables. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 2 Identify discrete probability distributions. 14) A 15) A 16) A 17) A 18) A 19) A Page 20
3 Graph discrete probability distributions. 20) (a) x P(x) 1 0 8 1 2 3 3 8 3 8 1 8 (b) 21) Page 21
22) 23) Page 22
24) (a) x P(x) 0 0.125 1 0.428 2 0.256 3 0.108 4 0.083 (b) 4 Compute and interpret the mean and standard deviation of a discrete random variable. 25) A 26) A 27) μ = x p(x) = 2(0.2) + 6(0.3) + 10(0.3) + 15(0.2) = 8.2 28) A 29) A 30) A 31) A 32) A 33) A 34) A 35) A 36) A 37) μ = 1.596; σ = 1.098 38) A 39) A 40) A 41) A 5 Interpret the mean of a discrete random variable as an expected value. 42) A 43) A 44) A 45) A 46) μ = xp(x) 3.6 Page 23
47) Let x = cost of fare paid by passenger. The probability distribution for x is: x $50 $490 x(p) 1/5 4/5 The expected cost is E(x) = μ = x p(x) = $50 1 5 + $490 4 5 = $402.00 Since the expected cost is more than the usual one-way air fare, the passenger should not opt to fly as a standby. 48) To determine the premium, the insurance agency must first determine the average loss paid on the sports car. Let x = amount paid on the sports car loss. The probability distribution for x is: x $24,500 $12,000 $5,750 $2,000 -$500 p(x) 0.001 0.01 0.05 0.10 0.839 Note: These losses paid have already considered the $500 deductible paid by the owner. The expected loss paid is: μ = x p(x) = $24,500(0.001) + $12,000(0.01) + $5,750(0.05) + $2,000(0.10) - $500(0.839) = $212.50 In order to average $500 profit per policy sold, the insurance company must charge an annual premium of $212.50 + $500 = $712.50. 49) A 50) A 51) A 6.2 The Binomial Probability Distribution 1 Determine whether a probability experiment is a binomial experiment. 1) binomial experiment 2) Not a binomial experiment. There are more than two outcomes. 3) Not a binomial experiment. There are more than two outcomes. 4) binomial experiment. 5) Not a binomial experiment. There are more than two outcomes. 6) binomial experiment. 7) binomial experiment. 8) Not a binomial experiment. There are more than two outcomes. 9) Not a binomial experiment. There are more than two outcomes. 10) Not a binomial experiment. The probability of success is not the same for each trial. 11) A 12) A 13) A 2 Compute probabilities of binomial experiments. 14) A 15) A 16) A 17) A 18) A 19) A 20) A 21) A 22) A 23) A 24) A 25) A 26) A 27) A 28) A Page 24