Chapter 3
Probability
True/False
1. A contingency table is a tabular summary of probabilities concerning two sets of complementary events.
Answer: True Difficulty: Medium
2. An event is a collection of sample space outcomes.
Answer: True Difficulty: Easy
3. Two events are independent if the probability of one event is influenced by whether or not the other event occurs.
Answer: False Difficulty: Medium
4. Mutually exclusive events have a nonempty intersection.
Answer: False Difficulty: Medium (REF)
5. A subjective probability is a probability assessment that is based on experience, intuitive judgment, or expertise.
Answer: True Difficulty: Medium
6. The probability of an event is the sum of the probabilities of the sample space outcomes that correspond to the event.
Answer: True Difficulty: Medium
7. If events A and B are mutually exclusive, then P() is always equal to zero.
Answer: True Difficulty: Hard (REF)
8. If events A and B are independent, then P(A|B) is always equal to zero.
Answer: False Difficulty: Medium (REF)
9. If events A and B are mutually exclusive, then P(AB) is always equal to zero.
Answer: True Difficulty: Easy
10. Events that have no sample space outcomes in common, and, therefore cannot occur simultaneously are referred to as independent events.
Answer: False Difficulty: Medium
Multiple Choice
11. Two mutually exclusive events having positive probabilities are ______dependent.
A) Always
B) Sometimes
C) Never
Answer: A Difficulty: Hard (REF)
12. ______is a measure of the chance that an uncertain event will occur.
A) Random experiment
B) Sample Space
C) Probability
D) A complement
E) A population
Answer: C Difficulty: Medium
13. A manager has just received the expense checks for six of her employees. She randomly distributes the checks to the six employees. What is the probability that exactly five of them will receive the correct checks (checks with the correct names).
A) 1
B) ½
C) 1/6
D) 0
E) 1/3
Answer: D Difficulty: Hard
14. In which of the following are the two events A and B, always independent?
A) A and B are mutually exclusive.
B) The probability of event A is not influenced by the probability of event B.
C) The intersection of A and B is zero.
D) P(A/B) = P(A).
E) B and D.
Answer: E Difficulty: Hard (REF)
15. If two events are independent, we can _____ their probabilities to determine the intersection probability.
A) Divide
B) Add
C) Multiply
D) Subtract
Answer: C Difficulty: Easy
16. Events that have no sample space outcomes in common, and therefore, cannot occur simultaneously are:
A) Independent
B) Mutually Exclusive
C) Intersections
D) Unions
Answer: B Difficulty: Medium
17. If events A and B are independent, then the probability of simultaneous occurrence of event A and event B can be found with:
A) P(A)·P(B)
B) P(A)·P()
C) P(B)·P()
D) All of the above are correct
Answer: D Difficulty: Hard (REF)
18. The set of all possible experimental outcomes is called a(n):
A) Sample space
B) Event
C) Experiment
D) Probability
Answer: A Difficulty: Easy
19. A(n) ______is the probability that one event will occur given that we know that another event already has occurred.
A) Sample space outcome
B) Subjective Probability
C) Complement of events
D) Long-run relative frequency
E) Conditional probability
Answer: E Difficulty: Medium
20. The ______of two events X and Y is another event that consists of the sample space outcomes belonging to either event X or event Y or both event X and Y.
A) Complement
B) Union
C) Intersection
D) Conditional probability
Answer: B Difficulty: Medium
21. If P(A) > 0 and P(B) > 0 and events A and B are independent, then:
A) P(A) = P(B)
B) P()=P(A)
C) P(AB) = 0
D) P(AB) = P(A) P(BA)
Answer: B Difficulty: Medium
22. P(AB) = P(A) + P(B) - P(AB) represents the formula for the
A) conditional probability
B) addition rule
C) addition rule for two mutually exclusive events
D) multiplication rule
Answer: B Difficulty: Medium
23. The management of a company believes that weather conditions significantly affect the level of demand for its product. 48 monthly sales reports are randomly selected. These monthly sales reports showed 15 months with high demand, 28 months with medium demand, and 5 months with low demand. 12 of the 15 months with high demand had favorable weather conditions. 14 of the 28 months with medium demand had favorable weather conditions. Only 1 of the 5 months with low demand had favorable weather conditions. What is the probability that weather conditions are poor, given that the demand is high?
A) .2
B) .5
C) .8
D) .25
E) .75
Answer: A Difficulty: Hard
24. The management believes that the weather conditions significantly impact the level of demand and the estimated probabilities of poor weather conditions given different levels of demand is presented below.
What is the probability of high demand given that the weather conditions are poor.
A) .06
B) .44
C) .1364
D) .12
E) .1818
Answer: C Difficulty: Hard
Use the following information to answer questions 25-26:
An automobile insurance company is in the process of reviewing its policies. Currently drivers under the age of 25 have to pay a premium. The company is considering increasing the value of the premium charged to drivers under 25. According to company records, 35% of the insured drivers are under the age of 25. The company records also show that 280 of the 700 insured drivers under the age of 25 had been involved in at least one automobile accident. On the other hand, only 130 of the 1300 insured drivers 25 years or older had been involved in at least one automobile accident.
25. An accident has just been reported. What is the probability that the insured driver is under the age of 25?
A) 35%
B) 20.5%
C) 14%
D) 68.3%
E) 40%
Answer: D Difficulty: Hard (AS)
26. What is the probability that an insured driver of any age will be involved in an accident?
A) 35%
B) 20.5%
C) 65%
D) 68.3%
E) 79.5%
Answer: B Difficulty: Hard (AS)
27. A pharmaceutical company manufacturing pregnancy test kits wants to determine the probability of a woman not being pregnant when the test results indicate pregnancy. It is estimated that the probability of pregnancy among potential users of the kit is 10%. According to the company laboratory test results 1 out of 100 non-pregnant women tested pregnant (false positive). On the other hand, 1 out of 200 pregnant women tested non-pregnant (false negative). A woman has just used the pregnancy test kit manufactured by the company and the results showed pregnancy. What is the probability that she is not pregnant?
A) 90%
B) 0.9%
C) 8.3%
D) 91.7%
E) 10.85%
Answer: C Difficulty: Hard
28. A pharmaceutical company manufacturing pregnancy test kits wants to determine the probability of a woman actually being pregnant when the test results indicate that she is not pregnant. It is estimated that the probability of pregnancy among potential users of the kit is 10%. According to the company laboratory test results 1 out of 100 non-pregnant women tested pregnant (false positive). On the other hand, 1 out of 200 pregnant women tested non-pregnant (false negative). A woman has just used the pregnancy test kit manufactured by the company and the results showed that she is not pregnant. What is the probability that she is pregnant?
A) 1%
B) 0.9%
C) 0.05%
D) 8.3%
E) 0.056%
Answer: E Difficulty: Hard
Fill-in-the-Blank
29. A(n) _____ is the set of all of the distinct possible outcomes of an experiment.
Answer: Sample Space Difficulty: Medium
30. The _____ of an event is a number that measures the likelihood that an event will occur when an experiment is carried out.
Answer: Probability Difficulty: Easy
31. When the probability of one event is influenced by whether or not another event occurs, the events are said to be _____.
Answer: Dependent Difficulty: Medium
32. A process of observation that has an uncertain outcome is referred to as a(n) _____.
Answer: Experiment Difficulty: Medium
33. When the probability of one event is not influenced by whether or not another event occurs, the events are said to be _____.
Answer: Independent Difficulty: Medium
34. A probability may be interpreted as a long run _____ frequency.
Answer: Relative Difficulty: Medium
35. If events A and B are independent, then P(A/B) is equal to _____.
Answer: P(A) Difficulty: Medium
36. The simultaneous occurrence of event A and B is represented by the notation: ______.
Answer: AB Difficulty: Easy
37. A(n) ______probability is a probability assessment that is based on experience, intuitive judgment, or expertise.
Answer: Subjective Difficulty: Medium
38. A(n) ______is a collection of sample space outcomes.
Answer: Event Difficulty: Easy
39. Probabilities must be assigned to experimental outcomes so that the probabilities of all the experimental outcomes must add up to ___.
Answer: 1 Difficulty: Easy
40. Probabilities must be assigned to experimental outcomes so that the probability assigned to each experimental outcome must be between ____ and ____ inclusive.
Answer: 0,1 Difficulty: Easy
41. The ______of event X consists of all sample space outcomes that do not correspond to the occurrence of event X.
Answer: Complement Difficulty: Easy
42. The ______of two events A and B is another event that consists of the sample space outcomes belonging to either event A or event B or both event A and B.
Answer: Union Difficulty: Easy
43. The ______of two events A and B is the event that consists of the sample space outcomes belonging to both event A and event B.
Answer: Intersection Difficulty: Easy
44. ______statistics is an area of statistics that uses Bayes' theorem to update prior belief about a probability or population parameter to a posterior belief.
Answer: Bayesian Difficulty: Medium
45. In the application of Bayes' theorem the sample information is combined with prior probabilities to obtain ______probabilities.
Answer: posterior Difficulty: Easy (REF)
Essay
46. What is the probability of rolling a seven with a pair of fair dice?
Answer: 1/6
Difficulty: Medium
47. What is the probability of rolling a value higher than eight with a pair of fair dice?
Answer: .2777
Difficulty: Medium
48. What is the probability that an even number appears on the toss of a die?
Answer: .5 Difficulty: Easy
49. What is the probability that a king appears in drawing a single card form a deck of 52 cards?
Answer: 1/13 Difficulty: Medium
50. If we consider the toss of four coins as an experiment, how many outcomes does the sample space consist of?
Answer: 16
Difficulty: Medium
51. What is the probability of at least one tail in the toss of three fair coins?
Answer: 7/8 Difficulty: Hard
52. A lot contains 12 items, and 4 are defective. If three items are drawn at random from the lot, what is the probability they are not defective?
Answer: .2545
Difficulty: Hard
53. A person is dealt 5 cards from a deck of 52 cards. What is the probability they are all clubs?
Answer: .0004951
Difficulty: Hard
54. A group has 12 men and 4 women. If 3 people are selected at random from the group, what is the probability that they are all men?
Answer: .392857
Difficulty: Hard
Use the following information to answer questions 55-57:
Container 1 has 8 items, 3 of which are defective. Container 2 has 5 items, 2 of which are defective. If one item is drawn from each container:
55. What is the probability that both items are not defective?
Answer: .375
Difficulty: Medium
56. What is the probability that the item from container one is defective and the item from container 2 is not defective?
Answer: .225
Difficulty: Hard
57. What is the probability that one of the items is defective?
Answer: .45
Difficulty: Hard
58. A coin is tossed 6 times. What is the probability that at least one head occurs?
Answer: 63/64
Difficulty: Medium
59. Suppose P(A) = .45, P(B) =.20, P(C) = .35, P() = .10, P() = .05, and P() = 0. What is P(E)?
Answer: .055
P(E) = (.45)(.10) + (.20)(.05) + (.35)(0) = .055
Difficulty: Hard
60. Suppose P(A) = .45, P(B) = .20, P(C) = .35, P() = .10, P() = .05, and P() = 0. What is P()?
Answer: .8182
Difficulty: Hard
61. Suppose P(A) = .45, P(B) = .20, P(C) = .35, P() = .10, P() = .05, and P() = 0. What is P()?
Answer: 1818
Difficulty: Hard
62. Suppose P(A) = .45, P(B) = .20, P(C) = .35, P() = .10, P() = .05, and P() = 0. What is P ()?
Answer: 0
Difficulty: Hard
63. Given the standard deck of cards, what is the probability of drawing a red card, given that it is a face card?
Answer: .5
Difficulty: Medium
64. Given a standard deck of cards, what is the probability of drawing a face card, given that it is a red card?
Answer: 3/13
Difficulty: Medium
65. A machine is made up of 3 components: an upper part, a mid part, and a lower part. The machine is then assembled. 5 percent of the upper parts are defective; 4 percent of the mid parts are defective; 1 percent of the lower parts are defective. What is the probability that a machine is non-defective?