CHAPTER
5
Discrete Random Variables and Their Probability Distributions 1. Explain the meaning of a random variable, a discrete random variable, and a continuous random variable. Give one example each of a discrete random variable and a continuous random variable. 2. Classify the following random variables as discrete or continuous. a. The number of tornadoes sighted in Texas during the month of April b. The amount of caffeine in a cup of coffee c. The body weight (in pounds) gained by a woman during pregnancy d. The number of female employees working for a company 3. Classify the following random variables as discrete or continuous. a. The number of phone calls received at a police station on a day b. The number of injuries occurring to children at a school during a month c. The time spent by a lawyer with a client d. The price of a football game ticket 4. Classify the following random variables as discrete or continuous. a. The number of doctors in a city b. The age of an antique wooden table c. The amount of milk in a gallon of milk d. The number of cars stopping at a gas station on a given day 5. Explain the meaning of the probability distribution of a discrete random variable. Give one example of such a probability distribution. What are the three ways to present the probability distribution of a discrete random variable? 6. Briefly explain the two characteristics (or conditions) of the probability distribution of a discrete random variable. 7.
The following table lists certain values of x and their probabilities. Verify whether each of the three divisions ( a. through c.) represents a valid probability distribution. Explain. a. x 0 1 2 3
b. P(x) .10 .22 .40 .28
x 2 3 4 5
c. P(x) .04 .37 .21 .12
x 7 8 9
P(x) .43 .68 –.11
8. The following table lists the probability distribution of the number of items produced per hour by a machine. x P(x) x P(x) 8 .04 12 .11 107
9 10 11
.17 .28 .33
13 14
.04 .03
a. Draw a probability histogram for this probability distribution. b. Find the probability that during a given hour this machine will produce: (i) exactly 10 items (ii) more than 11 items (iii) less than 12 items (iv) 9 to 12 items 9. The following table lists the probability distribution of the number of customers visiting a convenience store per 15 minutes. x 5 6 7 8 9 10
P(x) .12 .20 .27 .18 .15 .08
a. Draw a probability histogram for this probability distribution. b. Find the probability that during a given 15-minute period, the number of customers visiting this store will be: (i) exactly 8 (ii) more than 7 (iii) less than 8 (iv) 7 to 9 10. The following table gives the frequency distribution of the number of suits owned by all 100 managers of all companies in a city. Number of Suits 3 4 5 6 7 8
Number of Managers 5 11 31 28 16 9
a. Construct the probability distribution table for the number of suits owned by managers of these 100 companies. b. Let x denote the number of suits owned by a manager selected at random from these companies. Find the following probabilities. (i) P(x = 5) (iv) P(4 x 7) (ii) P(x 6) (v) P(x < 6) (iii) P(x 5) (vi) P(x > 5) 11. The following table gives the frequency distribution of the number of errors on each page of a book.
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Errors 0 1 2 3 4
Number of Pages 144 72 63 15 6
a. Construct the probability distribution table for the number of errors per page for this book. b. Let x denote the number of errors on a page selected at random from this book. Find the following probabilities. (i) P(x = 3) (iv) P(1 x 3) (ii) P(x 2) (v) P(x < 2) (iii) P(x 3) (vi) P(x > 2) 12. Five percent of all floppy diskettes made on a machine are defective. Let x be the number of defective floppy diskettes in a random sample of two floppy diskettes selected from the production line of this machine. a. Construct the probability distribution of x. b. Draw a tree diagram for this problem. 13. Ten percent of all families watch 60 minutes on CBS network. Let x be the number of families in a random sample of two families that watch 60 minutes. a. Construct the probability distribution of x. b. Draw a tree diagram for this problem. 14. In a group of 20 persons, 14 own domestic cars. Suppose you select two persons randomly from this group. Let x be the number of persons in this sample who own domestic cars. a. Construct the probability distribution of x. b. Draw a tree diagram for this problem. 15. In a group of 15 stockbrokers, six are women. Suppose you select two persons randomly from this group of 15 stockbrokers. Let x be the number of persons in this sample who are women. a. Construct the probability distribution of x. b. Draw a tree diagram for this problem.
Sections 5.3-5.4 16. The following table lists the probability distribution of the number of items produced per hour by a machine. x P(x) x P(x) 8 .04 12 .11 9 .17 13 .04 14 .03 10 .28 11 .33 Find the mean and standard deviation for this probability distribution. Give a brief interpretation of the value of the mean. 17. The following table lists the probability distribution of the number of customers visiting a convenience store per 15 minutes. 109
x 5 6 7
P(x) .12 .20 .27
x 8 9 10
P(x) .18 .15 .08
Find the mean and standard deviation for this probability distribution. Give a brief interpretation of the value of the mean. 18. The following table lists the probability distribution of the number of phone calls received per hour at an office of the Alcoholic Anonymous. x 3 4 5
P(x) .06 .11 .22
x 6 7 8
P(x) .26 .18 .17
Find the mean and standard deviation for this probability distribution. Give a brief interpretation of the value of the mean. 19. An instant lottery ticket costs $1. In a total of 10,000 tickets for this lottery, 1000 tickets contain a prize of $3 each, 100 tickets have a prize of $5 each, five tickets have a prize of $100 each, and one ticket has a prize of $1000. Let x be the random variable that denotes the net amount a player wins by playing this lottery. a. Write the probability distribution of x. b. Determine the mean and standard deviation of x. c. How will you interpret the value of the mean of x?
Section 5.5 20. Find the value of each of the following using the appropriate formula. a. 9! b. (13 – 8)! c. (7 – 0)! d. (4 – 4)! 21. Find the value of each of the following using the appropriate formula. a. 6! b. (22 – 15)! c. (5 – 0)! d. (7 – 7)! 22. A teacher selects two students at random from a group of 18 for being members of a committee. How many total combinations are possible? Use the appropriate formula. 23. A magician intends to select five items from a box that contains 20 items. In how many ways can he select five items from 20? Use the appropriate formula. 24. A student will select two topics at random from a list of 12 topics to write essays for a course. How many total combinations are possible? Use the appropriate formula. 25. Briefly define the following terms. a. a binomial experiment 110
b. a trial 26. What are the conditions that an experiment must satisfy to be a binomial experiment? 27. What are the parameters of the binomial probability distribution and what do they mean? 28. The binomial probability distribution is symmetric when p = .50, it is skewed to the right when p < .50, and it is skewed to the left when p > .50. Illustrate these three cases by writing three probability distributions and graphing them. Choose any values of n and p and use the table of binomial probabilities. 29. Six percent of floppy disks manufactured by a company are defective. Using the binomial formula, find the probability that in a random sample of 10 floppy disks selected from the production line of this company, exactly one will be defective. 30. The editor of a journal historically accepts 12 percent of articles submitted for publication. Using the binomial formula, find the probability that in a random sample of eight articles submitted to this journal, the editor will accept exactly three for publication. 31. Sixteen percent of adults contribute to charitable agencies on a regular basis. Using the binomial formula, find the probability that in a random sample of 12 adults, exactly five contribute to charitable agencies on a regular basis. 32. Sixty-four percent of all airplanes arriving at an airport are late. Using the binomial formula, find the probability that in a random sample of seven airplanes, exactly four will arrive late. 33. Ten percent of all households watch Monday Night Football on ABC network. Using the binomial probabilities table, find the probability that in a random sample of 14 households, at most five watch Monday Night Football on ABC network. 34. Forty percent of all families own two or more cars. Using the binomial probabilities table, find the probability that in a random sample of 16 families, at most four own two or more cars. 35. Sixty percent of adults are in favor of the death penalty for drug dealers. Using the binomial probabilities table, find the probability that in a random sample of nine adults, at least six will be in favor of the death penalty for drug dealers. 36. Seventy percent of adults are in favor of reducing defense expenditure and spending the same money on education and health care. Using the binomial probabilities table, find the probability that in a random sample of eight adults, at least five will be in favor of reducing defense expenditure and spending the same money on education and health care. 37. Twenty percent of drivers driving between 11 PM and 3 AM are drunken drivers. Using the binomial probabilities table, find the probability that in a random sample of 12 drivers driving between 11 PM and 3 AM, two to four will be drunken drivers. 38. Twenty-eight percent of adults contribute to charitable agencies on a regular basis. Let x be the number of adults in a random sample of 18 adults who contribute to charitable agencies on a regular basis. Find the mean and standard deviation of the probability distribution of x.
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39. Forty percent of all families own two or more cars. Let x be the number of families in a random sample of 15 families who own two or more cars. Find the mean and standard deviation of the probability distribution of x. 40. List the required conditions to apply the hypergeometric probability distribution. 41. You just purchased 50 bargain CD ROMs from a local office supply store. While trying to format the disks for storing data, you discover that four of them will not format and the rest will. By accident, your younger sister mixes the defective disks together with the good ones. You decide to select six from the batch of 50. What is the probability that two of these are defective and the other four are already formatted? 42. What are the required conditions to apply the Poisson probability distribution? What is the parameter of the Poisson distribution, and what does it mean? 43. On average, 6.2 customers visit a store per hour. Find the probability that during a given hour exactly nine customers will visit this store. Use the Poisson formula. 44. A fabric contains on average 2.2 defects per 200 yards. Find the probability that a given piece of 200 yards of this fabric will contain no defects. Use the Poisson formula. 45. On average, each page of a newspaper contains 2.5 spelling errors. Find the probability that a given page of this newspaper will contain exactly five spelling errors. Use the Poisson formula. 46. A mail order company makes an average of 11 sales per half hour. Find the probability that during a given half hour this company will make at most six sales. Use the Poisson probabilities table. 47. A university computer breaks down on average 2.1 times a month. Find the probability that during the next month this computer will break down at least six times. Use the Poisson probabilities table. 48. On average, the citizens report 1.5 burglary incidents to the police per day. Find the probability that on a given day, the citizens report more than four burglary incidents to this police station. Use the Poisson probabilities table. 49. An auto salesperson sells on average 4.2 cars per month. Find the probability that during a given month, the salesperson will sell less than five cars. Use the Poisson probabilities table.
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Chapter Review Questions 50.
One thousand batteries coming off a production line are tested for defects. The following table gives the frequency distributions of the number of defects observed. Number of Defects 0 1 2 3 4
Frequency 700 110 90 80 20
Suppose you choose a battery at random from the 1000 that were tested. The random variable x represents the number of defects that this battery has. What is the probability distribution of x? 51. A survey team is collecting data on sixty U.S. cities to determine how ―livable‖ they are. A city’s ―livability‖ is a measure of how much a person would like to live there if given the chance. Data are collected on, and random variables assigned to, the following properties: crime rate as a percentage of population amount of pollutants in the air, in parts per million number of businesses in the city employing more than five hundred people average temperature average humidity number of restaurants in the city professional sports teams that play home games in the city colleges and universities within one hour of the city limits Which of these properties can be represented by discrete random variables, and which can be represented by continuous random variables? 52.
A consulting company is currently developing separate software packages for 20 different insurance companies. Before any of this software can go out, the company must thoroughly test it for bugs in the program code. A major bug severely hinders the use of the software. The company collects data on each software package to isolate the number of major bugs discovered in testing. Although some of the data is lost, the company salvaged the following data: Number of Major Bugs 0 1 2
Probability .65 ? ?
Before the data were lost, the company calculated that the mean number of flaws for the 20 packages is .45. Complete the probability distribution. 53.
Employees at a particular company are eligible for performance raises at the end of each fiscal year. Management expresses raises as a percentage of salary. The percentage raise received by a randomly-selected employee has the following probability distribution.
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Raise (%) 3 4 5 6 7 8
Probability .02 .10 .28 .32 .20 .08
Find are the mean, variance, and standard deviation. 54. The saxophone section of a jazz band is to have four members—one alto sax player and three tenor sax players. The band director is choosing from two equally-talented alto sax players and six equally-talented tenor sax players. In how many ways can the band director select the four members of the saxophone section? 55. Sixty senior girls attend their high school prom. The prom committee must select eight of the girls to be part of the prom ―court‖. In how many ways can the committee make the selection? 56. Heavy flooding has caused the Mississippi River to rise past flood stage in a certain community, forcing the army corps of engineers to construct levees along the river. Each halfmile section of the levees has a 75% chance of holding without significant leakage, a 20% chance of having significant leakage, and a 5% chance of breaking altogether. The corps of engineers has determined that if 4 or more sections of the levees break, then there will be widespread flooding. You want to determine the probability that 3 or fewer sections break. Is this a binomial experiment? Explain. 57. A doctor is trying an experimental drug with 20 AIDS patients. The doctor feels that the probability that the drug will improve the T-cell count of a randomly-selected patient is .20. The random variable z represents the number of patients whose T-cell counts improve while they are on the medication. Assuming the doctor’s assumption is accurate, what is the function P(z)? Also, compute the mean and variance of z. 58. You are choosing between two games. In one game you flip a fair coin 10 times, and you win if you get exactly 4 tails. In the second game, you flip a weighted coin. Heads comes up 80% of the time, and tails comes up 20% of the time. In the second game, however, you only need to get one tail out of 10 flips. In which game do you have a better chance of winning? 59. A bag contains 40 black marbles and 20 red marbles. You select eight of them, one right after the other. What is the probability that five of the marbles you selected are black and three of them are red? 60. In a certain community, an average of six people dies each year in auto accidents. Assuming that x, the number of people that die in auto accidents in a given year, has a Poisson distribution with = 6, compute the mean and variance of x. What is P()?
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Problem and Essay Solutions 1. A random variable is a quantitative variable with values that vary according to the rules of probability. The values of a discrete random variable are integer or whole numbers. The values of a continuous random variable can be any value within the interval of a number line. An example of a discrete random variable is the number of cars passing through a toll gate in a given period of time. An example of a continuous random variable is the distance traveled by cars on the highway in one minute. 2. a. discrete c. continuous b. continuous d. discrete 3. a. discrete b. discrete
c. continuous d. continuous
4. a. discrete b. continuous
c. continuous d. discrete
5. The probability distribution of a discrete random variable gives the probability that the random variable will take on each of its possible values. The number of customer complaints received each day by a newspaper publisher trying to deliver its paper is an example. You could use a table of values, a histogram, and an ogive to display the distribution. 6. The two characteristics of the probability distribution of a discrete random variable are: a. The probability that a random variable x assumes a single value is in the range zero to 1. The mathematical expression is 0 P(x) 1. b. The sum of the probabilities of all values that a discrete random variable x can assume is 1.0. The mathematical expression is P(x) = 1. 7. a. Yes. The probabilities are valid and sum to 1. b. No. The probabilities are valid but do not sum to 1. c. No. The probabilities sum to 1, but –.11 is not a valid probability value. 8. a. Probability Histogram 0.35 0.3
P(x )
0.25 0.2 0.15 0.1 0.05 0 8
9
10
11 x
b. (i) (ii) (iii) (iv)
P(x = 10) = .28 P(x > 11) = .11 + .04 + .03 = .18 P(x < 12) = 1 – (.11 + .04 + .03) = .82 P(9 x 12) = .17 + .28 + .33 + .11 = .89
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12
13
14
9. a. Probability Histogram 0.3 0.25
P(x )
0.2 0.15 0.1 0.05 0 5
6
7
8 x
b. (i) (ii) (iii) (iv)
P(x = 8) = .18 P(x > 7) = .18 + .15 + .08 = .41 P(x < 8) = 1 – (.18 + .15 + .08) = .59 P(7 x 9) = .27 + .18 + .15 = .60
10. a. x 3 4 5 6 7 8 b. (i) (ii) (iii) (iv) (v) (vi)
P(x) .05 .11 .31 .28 .16 .09
P(x = 5) = .31 P(x 6) = .28 + .16 + .09 = .53 P(x 5) = .31 + .11 + .05 = .47 P(4 x 7) = .11 + .31 + .28 + .16 = .86 P(x < 6) = .31 + .11 + .05 = .47 P(x > 5) = .28 + .16 + .09 = .53
11. a. x 0 1 2 3 4 b. (i) (ii) (iii) (iv) (v) (vi)
P(x) .48 .24 .21 .05 .02
P(x = 3) = .05 P(x 2) = .21 + .05 + .02 = .28 P(x 3) = 1 – .02 = .98 P(1 x 3) = .24 + .21 + .05 = .50 P(x < 2) = .48 + .24 = .72 P(x > 2) = .05 + .02 = .07
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9
10
12. a. x 0 1 2
P(x) .9025 .0950 .0025
b. D = Defect, ND = No Defect D D ND D ND ND 13. a. x 0 1 2
P(x) .81 .18 .01
b. W = Watch, DW = Don’t Watch W W DW W DW DW 14. a. x 0 1 2
P(x) .0789 .4422 .4789
b. O = Own, DO = Don’t Own O O DO O DO DO
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15. a. x 0 1 2
P(x) .3429 .5142 .1429
b. W = Woman, M = Man W W M W M M 16. Mean = 10.54, Standard deviation = 1.284 17. Mean = 7.28, Standard deviation = 1.450 18. Mean = 5.90, Standard deviation = 1.432 19. a. x –1 2 4 99 999
P(x) .8894 .1000 .0100 .0005 .0001
xP(x) –.8894 .2000 .0400 .0495 .0999
b. Mean = –$.50, Standard deviation = $10.204 c. One can expect to lose $.50 per lottery ticket playing it continuously. 20. a. b. c. d.
9! = 9 8 7 6 5 4 3 2 1 = 362,880 (13 – 8)! = 5! = 5 4 3 2 1 = 120 (7 – 0)! = 7! = 7 6 5 4 3 2 1 = 5040 (4 – 4)! = 0! = 1
21. a. b. c. d.
6! = 6 5 4 3 2 1 = 720 (22 – 15)! = 7! = 7 6 5 4 3 2 1 = 5040 (5 – 0)! = 5! = 5 4 3 2 1 = 120 (7 – 7)! = 0! = 1
22.
18C2
=
18! = 153 2!(18 2)!
23.
20C5
=
20! = 15,504 5!(20 5)!
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24.
12C2
=
12! = 66 2!(12 2)!
25. a. A binomial experiment consists of repeating a basic experiment that can have only two possible outcomes a certain number of times. b. A trial is a basic experiment that has only two possible outcomes, success or failure. 26. A binomial experiment must satisfy the following conditions: The experiment is performed a fixed number of times or trials. Each trial is identical and has two possible outcomes, either a success or a failure. The trials are independent of one another. 27. The parameters are n and p, where n is the number of trials and p is the probability of a success. 28. p = .2 .26214 .39322 .24576 .08192 .01536 .00154 .00006
x 0 1 2 3 4 5 6
p = .5 .01563 .09375 .23438 .31250 .23438 .09375 .01563
p = .7 .00073 .01021 .05954 .18522 .32414 .30253 .11765
p = .2 0.5000
P(x )
0.4000 0.3000 0.2000 0.1000 0.0000 0
1
2
3
4
5
6
4
5
6
x
p = .5 0.3500 0.3000
P(x )
0.2500 0.2000 0.1500 0.1000 0.0500 0.0000 0
1
2
3 x
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p = .7 0.3500 0.3000
P(x )
0.2500 0.2000 0.1500 0.1000 0.0500 0.0000 0
1
2
3
4
5
6
x
29. P(1) =
10! .061.94101 = .3438 1!(10 1)!
30. P(3) =
8! .12 3.8883 = .0511 3!(8 3)!
31. P(5) =
12! .16 5.84125 = .0245 5!(12 5)!
32. P(4) =
7! .64 4.36 74 = .2740 4!(7 4)!
33. .9986 34. .1665 35. .4826 36. .8059 37. .6526 38. Mean = (18)(.28) = 5.04, Standard deviation = 39. Mean = (15)(.4) = 6, Standard deviation =
(18)(. 28)(. 72) = 1.905
(15)(. 4)(. 6) = 1.897
40. The requirements for applying the hypergeometric distribution are as follows: 1) Trials are not independent. 2) The sample is drawn without replacement. 3) The population is finite. 41. N = 50, r = 46, n = 6, x = 4 Applying the hypergeometric distribution:
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46! 4! C C 4! ( 46 4)! 2! ( 4 2)! (163,185)(6) P( x 4) 46 4 4 2 .06162 50! 15,890,700 50 C 6 6! (50 6)!
42. The Poisson distribution is a limiting form of the binomial distribution. It is applicable to those situations when the number of trials is large and the probability of a success is very small. The parameter, , indicates the expected number of occurrences of the event. One difference between the binomial and the Poisson distribution is that the number of trials is known when the binomial is applied, whereas the number of trials is usually unknown when the Poisson is applied. 43. P(k = 9; 6.2) = e 6.2 44. P(k = 0; 2.2) = e 2.2
45. P(k = 5; 2.5) = e 2.5
6.2 9 = .0757 9! 2.2 0 = .1108 0! 2.5 5 = .0668 5!
46. P(k 6; 11) = .0786 47. P(k 6; 2.1) = .0146 + .0044 + .0011 + .0003 + .0001 = .0205 48. P(k > 4; 1.5) = .0141 + .0035 + .0008 + .0001 = .0185 49. (k < 5; 4.2) = .0150 + .0630 + .1323 + .1852 + .1944 = .5899 50. # Defects 0 1 2 3 4
Probability .70 .11 .09 .08 .02
51. Discrete: number of businesses, number of restaurants, pro sports teams, colleges and universities Continuous: crime rate, pollutants, average temp, average humidity 52.
We know that P(0) + P(1) + P(2) = 1, so P(1) + P(2) = .35, or P(1) = .35 – P(2). Since the mean is .45, we have .45 = P(1) + 2P(2) = .35 – P(2) + 2P(2) = .35 + P(2). Therefore P(2) = .10, and P(1) = .25. The completed probability distribution is: Number of Major Bugs 0 1 2 121
Probability .65 .25 .10
53. = 3 .02 + 4 .10 + 5 .28 + 6 .32 + 7 .20 + 8 .08 = 5.82% 2 = 9 .02 + 16 .10 + 25 .28 + 36 .32 + 49 .20 + 64 .08 – (5.82)2 = 1.3476 = 1.3476 = 1.1609 54. There are two choices for the alto sax and 6C3 = 20 combinations of tenor sax players, for a total of 2 20 = 40 possible combinations. 55. There are 60C8 = 2,558,620,845 possible ways. 56. This is not a binomial experiment. For one thing, there are more than two possible outcomes for each trial. Also, it is doubtful that the trials are independent since if one part of a levee breaks, more strain will result on other parts of the levee. 57. P(z) will be binomial with n = 20, p = .20, and q = .80. = np = (20)(.2) = 4 2 = npq = (20)(.2)(.8) = 3.2 58. In the first game, the probability of winning is (10!/4!6!)(.50)10 = .2051 In the second game, the probability of winning is (10!/1!9!)(.20)(.80)9 = .2684 Therefore, the probability of winning is higher in the second game. 59. N = 60, r = 40, n = 8, x = 5 Applying the hypergeometric distribution: 40! 20! C C 5! ( 40 5)! 3! ( 20 3)! (658,008)(1140) P( x 5) 40 5 20 3 .2932 60! 2,558,620,845 60 C8 8! (60 8)! 60. Since we have a Poisson distribution, = mean = variance = 6. P(k = 6; 6) = e 6
66 = .1606 6!
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