Probability 1. You randomly select two households and observe whether or not they own VCRs. a. How many total outcomes are possible for this experiment? Draw a tree diagram and a Venn diagram for this experiment. b. List all the outcomes included in each of the following events and indicate which are simple and which are compound events? (i) at least one household owns a VCR (ii) at most one household owns a VCR (iii) both households own a VCR (iv) the first household owns a VCR and the second household does not own a VCR 2. A box contains a few pink and a few black marbles. You randomly draw two marbles from this box and the color of these marbles is observed. a. How many total outcomes are possible for this experiment? Draw a tree diagram and a Venn diagram for this experiment. b. List all the outcomes included in each of the following events and indicate which are simple and which are compound events? (i) at most one marble is pink (ii) not more than one marble is black (iii) none of the marbles is pink (iv) the first marble is black and the second marble is pink 3. What are the two properties of probability? Give a brief explanation of each of those. 4. In a group of 200 households, 124 own telephone answering machines. Select randomly one household from this group. What is the probability that this household owns a telephone answering machine? 5. You toss an unbalanced 900 times and observe a head 540 times. What is the approximate probability of observing a head for this coin? 6. A multiple-choice question on a test has five answers. If a student randomly selects one answer from these five, what is the probability that the selected answer is correct? 7. In a total of 400 employees of a company, 280 are nonsmokers. Management selects randomly one employee from this company. What is the probability that this employee is a nonsmoker? 8. Of the 1,200 babies born at a hospital during the past five years, 660 were girls. What is the approximate probability that the next baby born at this hospital will be a girl? 9.
The following table gives the two-way classification of 400 students based on sex and whether or not they work while being full-time students. Male
Work 120 79
Do Not Work 60
Female
130
90
a. Select one student randomly from this group of 400 students. What is the probability that this student: (i) does not work (ii) is a female (iii) does not work given he is a male (iv) is a female given she works b. Are the events “male” and “do not work” mutually exclusive? Explain why or why not. c. Are the events “female” and “do not work” independent? Explain why or why not. d. What is the complementary event of the event “do not work”? What is the probability of this complementary event? 10. An independent research team inspects 300 batteries manufactured by two companies for being good or defective. The following table gives the two-way classification of these 300 batteries. Company A Company B
Good 140 130
Defective 10 20
a. The team selects one battery randomly from these 300 batteries. Find the probability that this battery: (i) is manufactured by company B (ii) is defective (iii) is good given that it is manufactured by company B (iv) is manufactured by company A given that it is defective b. Are the events “company A” and “defective” mutually exclusive? Explain why or why not. c. Are the events “good” and “company A” independent? Explain why or why not. d. What is the complementary event of the event “defective”? What is the probability of this complementary event?
11. There are a total of 300 professors at a university. Of them, 75 are female and 90 are professors in the social sciences. Of the 75 females, 30 are professors in social sciences. Are the events “female” and “professor in social sciences” independent? Are they mutually exclusive? Explain why or why not? 12. There are a total of 40 students in a class. Of them, 18 are male and 14 are seniors. Of the 18 males, six are seniors. Are the events “male” and “senior” independent? Are they mutually exclusive? Explain why or why not. 13. Let A be the event that a randomly selected family owns a house. The probability of event A is .68. What is the complementary event of A, and what is its probability? 14. Let A be the event that a randomly selected employee of a company is in favor of labor unions. The probability of event A is .56. What is the complementary event of A, and what is its probability? 15. The following table gives a two-way classification of all employees of a company based on their sex and whether or not they are college graduates.
Sex Male Female
College Graduate Yes No 35 50 25 40
Management selects one employee randomly from the company. Find the probabilities: a. P(female and college graduate) b. P(male and not a college graduate) 16. A medical research unit asks 500 people whether or not they visited their physicians’ offices during the last year. The following table gives a two-way classification of their responses. Sex Male Female
Visited Physician’s Office Last Year Yes No 210 90 160 40
If the unit randomly selects one person from this group, find the following probabilities. a. P(male and visited his physician’s office) b. P(female and did not visit her physician’s office) 17.
The following table gives a two-way classification of 1000 couples based on whether one or both spouses work and whether or not they have children. Work Status Both Spouses Work Only One Spouse Works
Have Children Yes No 140 260 380 220
Select one couple randomly from these 1000 couples and find the following probabilities. a. P(both spouses work and have no children) b. P(only one spouse works and have children) 18. A university has 250 faculty members. Of them, 50 are female. You select two faculty members at random from these 250 faculty members. Find the probability that both of them are females. 19. An insurance company isolates a group of 100 car owners, 40 own foreign cars. The company selects two persons at random from these 100 persons. Find the probability that neither of them owns a foreign car. 20. Forty percent of banks did not earn profits during 2003. The SEC selects two banks at random. Find the probability that neither of them earned profits in 2003. 21. The probability that a person has blood type A is .30. The Red Cross selects two persons at random. Find the probability that the first person has blood type A and the second person does not have blood type A. 22. The probability that a person drinks at least five cups of coffee a day is .25, and the probability that a person has a high blood pressure is .10. Assuming that these two events are independent, find the probability that a person selected at random drinks less than five cups of coffee a day and has a high blood pressure.
23. The probability that a family owns a house is .65 and that a family owns a camcorder is .12. Assuming that these two events are independent, find the probability that a family selected at random owns a house but does not own a camcorder. 24. The probability that a faculty member at a large university is a female is .30. The joint probability that a faculty member is a female and holds a doctoral degree is .24. Find the probability that a randomly selected faculty member from this university holds a doctoral degree given she is a female. 25. The probability that a family owns a VCR is .60. The joint probability that a family owns a VCR and a camcorder is .42. Find the probability that a randomly selected family owns a camcorder given that this family owns a VCR. 26. How many different outcomes are possible for five tosses of a coin? 27. The provost decides to form a committee by selecting one of 10 professors, one of 20 students, and one of six administrators. How many of outcomes are possible? 28. The following table gives the two-way classification of 400 students based on whether or not they work while being full-time students. Male Female
Work 120 130
Do Not Work 60 90
You select one student at random from these 400 students. Find the following probabilities. a. P(female or does not work) b. P(works or male) 29. The following table gives a two-way classification of all employees of a company based on their sex and whether or not they are college graduates. Sex Male Female
College Graduate Yes No 35 50 25 40
Management selects one employee randomly from the company. Find the following probabilities. a. P(male or a college graduate) b. P(female or not a college graduate) 30. A medical research team asks 500 people whether or not they visited their physicians’ offices during the last year. The following table gives a two-way classification of the responses. Sex Male Female
Visited Physician’s Office Last Year Yes No 210 90 160 40
The team randomly selects one person from this group. Find the following probabilities.
a. P(female or visited her physician’s office) b. P(male or did not visit his physician’s office) 31. The following table gives a two-way classification of 1000 couples based on whether one or both spouses work and whether or not they have children. Work Status Both Spouses Work Only One Spouse Works
Have Children Yes No 140 260 380 220
You select one couple randomly from among these 1000 couples. Find the following probabilities. a. P(both spouses work or have no children) b. P(only one spouse works or have children) 32. The probability that a lawyer is a female is .28, that a lawyer has a type A personality is .64, and that a lawyer is a female and has a type A personality is .10. Find the probability that a randomly selected lawyer is a female or has a type A personality. 33. Suppose 15% of all National Basketball Association players are over 7 feet tall, 68% weigh over 200 pounds, and 8% are over 7 feet tall and weigh over 200 pounds. If one of the NBA players is selected at random, find the probability that this player is either over 7 feet tall or weighs over 200 pounds. 34. A large company has a total of 300 female employees. Of them, 15 have a Master’s degree as the highest degree, 120 have a Bachelor’s degree, 140 have a high school diploma, and 25 have less than a high school diploma. Management selects randomly one female employee from this company. Find the probability that this employee has a Master’s degree or a high school diploma as the highest degree. 35. The probability that a person drinks at least five cups of coffee a day is .25 and the probability that a person has a high blood pressure is .10. Assuming that these two events are independent, find the probability that a person selected at random drinks less than five cups of coffee a day or has high blood pressure. 36. The probability that a family owns a house is .65 and that a family owns a camcorder is .12. Assuming that these two events are independent, find the probability that a family selected at random owns a house or does not own a camcorder. 37. A die with six sides is loaded so that the number two is twice as likely to be rolled as the numbers one, three, four, or six, and four times as likely to be rolled as the number five. Build a table that shows the probability of rolling each number, one through six. 38. A college student is planning to apply for a school grant. A friend of hers who works for the company that is offering the grant has told her that if she takes calculus in her freshman year and gets an A or B, she will be very strongly considered for the grant, but if she gets a C, D, or F, she will probably not be considered. If she doesn’t take calculus at all, she will remain a candidate. She estimates the following probabilities: Probability of getting the grant if she takes calculus and gets A or B: .80 Probability of getting the grant if she takes calculus and gets C, D, or F: .10 Probability of getting the grant if calculus not taken: .50
Probability of getting A or B in calculus: .60 What is the probability that she will get the grant if she takes calculus? What should she do to maximize the probability that she will get the grant? 39. Forty professors of sciences and liberal arts are asked where they would prefer to vacation if given the choice. Their grouped responses appear in the following table. Sciences Liberal Arts
Caribbean 8 10
Pacific Islands 4 5
Europe 3 5
Other 3 2
You choose one of the forty professors at random. What is the probability that the professor would choose the Caribbean? What is the probability that the professor is a sciences professor? Suppose the professor is a liberal arts professor. What is the probability that he/she would choose Europe? 40. You are watching a casino game where the player and a casino employee both roll a pair of fair dice. If the player’s roll is higher than the employee’s roll, the player wins. If the rolls are the same or the employee’s roll is higher than the player’s, the player loses. The casino employee rolls a six and a five for a total of 11. What is the probability that the player will win? 41. A pollster asks 50 residents of a certain community whether or not they favor term limits for Congressmen. Twenty of the residents are males who favor term limits, and six of the residents are females who do not favor term limits. The probability that a resident, selected at random, favors term limits is .70. How many residents are male and do not favor term limits? 42. Two types of computers (Types A and B) are used at a certain company that uses 200 computers in all. The users of each computer indicate whether they experience performance problems with their computers more than once per week. Fifty of the computers of Type A experience problems more than once per week. Seventy-five of the computers of Type B do not experience problems more than once per week. There are 120 computers of Type B at the company. Are the events “Type A computer” and “experience problems more than once per week” independent? Explain. 43. Can two events which are complementary be mutually exclusive? Can they be independent? Explain. 44. A statistician determines that a certain baseball player (Player A) hits a home run in 30% of the games in which he plays. The statistician also finds that if Player A hits a home run in a game, his team wins the game 80% of the time. What is the probability that the player hits a home run and his team does not win the game? 45. Part of a life insurance underwriter’s job is to decide whether a person who applies for life insurance is an insurable risk; that is, an underwriter decides whether the person who is applying for insurance will live long enough to make the policy profitable. If the person buys the insurance and dies soon after, then the insurance company will pay out a large sum of money without having collected very much money in premiums from the person. So an underwriter is making a judgment as to whether the person is going to die soon. What concept of probability does an underwriter use?
46. You toss a coin, then a roll a die, and then toss the coin again. You are to draw a tree diagram for this experiment. Each “leaf” of the tree will represent a possible outcome for the experiment. How many “leaves” will the tree have?
Solutions 1. Let: V = a household owns a VCR N = a household does not own a VCR a. Four possible outcomes: VV, VN, NV, NN V
S
V
VV
VN
NV
NN
PP
PB
BP
BB
N
V N N b. (i) (ii) (iii) (iv)
VN, NV, VV; a compound event NN, VN, NV; a compound event VV; a simple event VN; a simple event
2. Let: P = marble selected is pink B = marble selected is black a. Four possible outcomes: PP, PB, BP, BB P P
S
B
P B B b. (i) (ii) (iii) (iv)
PB, BP, BB; a compound event PP, PB, BP; a compound event BB; a simple event BP; a simple event
3. The two properties of probability are: (i) the probability of an event lies in the range zero to 1 (ii) the sum of the probabilities of all outcomes for an experiment is 1 4. 124 200 = .62 5. 540 900 = .60 6. 1 5 = .20 7. 280 400 = .70
8. 660 1,200 = .55 9. a. (i) 150 400 = .375 (ii) 220 400 = .550 (iii) 60 180 = .333 (iv) 130 250 = .520 b. The events are not mutually exclusive because they can happen together. c. P(female) = .550, P(femaledoes not work) = .600. Hence, these two events are not independent. d. The complementary event is “work” and its probability is .625 10. a. (i) 150 300 = .500 (ii) 30 300 = .100 (iii) 130 150 = .867 (iv) 10 30 = .333 b. Not mutually exclusive since they can happen together c. P(good) = .900, P(goodcompany A) = .933. Hence, these events are not independent. d. The complementary event is “good” and its probability is .900 11. P(Female) = 75 300 = .25, P(FemaleProfessor in social sciences) = 30 90 = .333. Hence, these events are not independent. The events are not mutually exclusive because there are 30 female professors in social sciences (joint occurrence). 12. P(Male) = 18 40 = .45, P(MaleSenior) = 6 14 = .43. Therefore, these events are not independent. The events are not mutually exclusive because there are six male seniors (joint occurrence). 13. Complementary event: “does not own a house” P(not A) = 1 – .68 = .32 14. Complementary event: “not in favor of labor unions” P(not A) = 1 – .56 = .44 15. a. 25 150 = .167 b. 50 150 = .333 16. a. 210 500 = .420 b. 40 500 = .080 a. 260 1,000 = .260 b. 280 1,000 = .380 18. (50 250)(49 249) = .0392
17.
19. (60 100)(59 99) = .358 20. .4 .4 = .16 21. .3 .7 = .21
22. (1 – .25) .10 = .075 23. .65 (1 – .12) = .572 24. .24 .30 = .800 25. .42 .6 = .700 26. 25 = 32 27. 10 20 6 = 1,200 28. a. (220 400) + (150 400) – (90 400) = .700 b. (250 400) + (180 400) – (120 400) = .775 29. a. (85 150) + (60 150) – (35 150) = .733 b. (65 150) + (90 150) – (40 150) = .767 30. a. (200 500) + (370 500) – (160 500) = .820 b. (300 500) + (130 500) – (90 500) = .680 31. a. (400 1,000) + (480 1,000) – (260 1,000) = .620 b. (600 1,000) + (520 1,000) – (380 1,000) = .740 32. .28 + .64 – .10 = .82 33. .15 + .68 – .08 = .75 34. (15 300) + (140 300) = .517 35. (1 – .25) + .10 – (1 – .25)(.10) = .775 36. .65 + (1 – .12) – (1 – .12)(.65) = .958 37. P(2) + P(2)/4 + 4 P(2)/2 = 1; P(2) = 4/13. Therefore, the complete table is as follows: Number Probability
1 2/13
2 4/13
3 2/13
4 2/13
5 1/13
6 2/13
38. If she takes calculus, her probability of getting the grant will be: P(A or B in calculus and gets grant) + P(C, D, or F in calculus and gets grant) = P(A or B)P(gets grantA or B) + P(C, D, or F)P(gets grantC, D, or F) = (.60)(.80) + (.40)(.10) = .48 + .04 = .52 > .50 (probability of getting grant if calculus not taken) So if she takes calculus, her probability of getting the grant will be higher than if she doesn’t take it. Therefore, she should take calculus. 39. 18 professors chose the Caribbean, so the first (marginal) probability is 18 40 = .45 18 of the professors are science professors, so the second (marginal) probability is also .45 22 of the professors are in liberal arts, and five of them chose Europe, so the third (conditional) probability is 5 22 = .227 40. The player will win only if he/she rolls a twelve, for a probability of 1/36. 41. The number of residents that favor term limits is .70 50 = 35. Twenty of these are males, so 15 are females. Six residents are female and do not favor term limits, so 50 – 20 – 15 – 6 = 9 residents are males who do not favor term limits. 42. P(Type B computer) = 120 200 = .60, so P(Type A computer) = .40 There are 120 Type B computers, and 75 of them do not experience any problems, so 45 of them do. That makes 95 computers that experience problems. Therefore, P(Type A computerexperience problems) = 50 95 = .526 .40. So the two events are not independent. 43. Complementary events are, by definition, mutually exclusive. For events A and B to be independent, we must have P(A) = P(AB) = P(A and B) P(B). But if A and B are complementary, then P(A and B) = 0, so we cannot have P(A) = P(AB). Therefore, complementary events are never independent. 44. It is given that P(home run) = .30 and P(win gamehome run) = .80. Therefore, P(lose gamehome run) = .20, and P(home run and lose game) = P(lose gamehome run)P(home run) = .06. 45. This is a subjective probability because the probability of death is a judgment call. 46.
Since each “leaf” represents an outcome, there will be 2 6 2 = 24 “leaves” depicted on the tree diagram.