112年 - SOCIETY OF ACTUARIES_EXAM P PROBABILITY_EXAM P SAMPLE QUESTIONS 1-50#119899

科目：Exam P：Probability機率 | 年份：112年 | 選擇題數：50 | 申論題數：0

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所屬科目：Exam P：Probability機率

選擇題 (50)

1. A survey of a group’s viewing habits over the last year revealed the following
information:
(i) 28% watched gymnastics
(ii) 29% watched baseball
(iii) 19% watched soccer
(iv) 14% watched gymnastics and baseball
(v) 12% watched baseball and soccer
(vi) 10% watched gymnastics and soccer
(vii) 8% watched all three sports.
Calculate the percentage of the group that watched none of the three sports during the last year.(A) 24%(B) 36%(C) 41%(D) 52%(E) 60%
2. The probability that a visit to a primary care physician’s (PCP) office results in neither lab work nor referral to a specialist is 35%. Of those coming to a PCP’s office, 30% are referred to specialists and 40% require lab work.
Calculate the probability that a visit to a PCP’s office results in both lab work and referral to a specialist. (A) 0.05 (B) 0.12 (C) 0.18 (D) 0.25 (E) 0.35
3. You are given P[A∪B]=0.7 and P[A∪B′]=0.9 . Calculate P[A]. (A) 0.2 (B) 0.3 (C) 0.4 (D) 0.6 (E) 0.8
4. An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is 0.44.
Calculate the number of blue balls in the second urn. (A) 4 (B) 20 (C) 24 (D) 44 (E) 64
5. An auto insurance company has 10,000 policyholders. Each policyholder is classified as
(i) young or old；
(ii) male or female； and
(iii) married or single.
Of these policyholders, 3000 are young, 4600 are male, and 7000 are married. The policyholders can also be classified as 1320 young males, 3010 married males, and 1400 young married persons. Finally, 600 of the policyholders are young married males.
Calculate the number of the company’s policyholders who are young, female, and single.(A) 280(B) 423(C) 486(D) 880(E) 896
6. A public health researcher examines the medical records of a group of 937 men who died in 1999 and discovers that 210 of the men died from causes related to heart disease. Moreover, 312 of the 937 men had at least one parent who suffered from heart disease, and, of these 312 men, 102 died from causes related to heart disease.
Calculate the probability that a man randomly selected from this group died of causes related to heart disease, given that neither of his parents suffered from heart disease. (A) 0.115 (B) 0.173 (C) 0.224 (D) 0.327 (E) 0.514
7. An insurance company estimates that 40% of policyholders who have only an auto policy will renew next year and 60% of policyholders who have only a homeowners policy will renew next year. The company estimates that 80% of policyholders who have both an auto policy and a homeowners policy will renew at least one of those policies next year.

Company records show that 65% of policyholders have an auto policy, 50% of policyholders have a homeowners policy, and 15% of policyholders have both an auto policy and a homeowners policy.

Using the company’s estimates, calculate the percentage of policyholders that will renew at least one policy next year. (A) 20% (B) 29% (C) 41% (D) 53% (E) 70%
8. Among a large group of patients recovering from shoulder injuries, it is found that 22% visit both a physical therapist and a chiropractor, whereas 12% visit neither of these. The probability that a patient visits a chiropractor exceeds by 0.14 the probability that a patient visits a physical therapist.
Calculate the probability that a randomly chosen member of this group visits a physical therapist. (A) 0.26 (B) 0.38 (C) 0.40 (D) 0.48 (E) 0.62
9. An insurance company examines its pool of auto insurance customers and gathers the following information:
(i) All customers insure at least one car.
(ii) 70% of the customers insure more than one car.
(iii) 20% of the customers insure a sports car.
(iv) Of those customers who insure more than one car, 15% insure a sports car.
Calculate the probability that a randomly selected customer insures exactly one car and that car is not a sports car.(A) 0.13(B) 0.21(C) 0.24(D) 0.25(E) 0.30
10. An actuary studying the insurance preferences of automobile owners makes the following
conclusions:
(i) An automobile owner is twice as likely to purchase collision coverage as disability coverage.
(ii) The event that an automobile owner purchases collision coverage is independent of the event that he or she purchases disability coverage.
(iii) The probability that an automobile owner purchases both collision and disability coverages is 0.15.

Calculate the probability that an automobile owner purchases neither collision nor disability coverage.(A) 0.18(B) 0.33(C) 0.48(D) 0.67(E) 0.82
11. A doctor is studying the relationship between blood pressure and heartbeat abnormalities in her patients. She tests a random sample of her patients and notes their blood pressures (high, low, or normal) and their heartbeats (regular or irregular). She finds that:
(i) 14% have high blood pressure.
(ii) 22% have low blood pressure.
(iii) 15% have an irregular heartbeat.
(iv) Of those with an irregular heartbeat, one-third have high blood pressure.
(v) Of those with normal blood pressure, one-eighth have an irregular heartbeat.
Calculate the portion of the patients selected who have a regular heartbeat and low blood pressure.(A) 2%(B) 5%(C) 8%(D) 9%(E) 20%
12. An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman in the population has only this risk factor (and no others). For any two of the three factors, the probability is 0.12 that she has exactly these two risk factors (but not the other). The probability that a woman has all three risk factors, given that she has A and B, is 1/3.
Calculate the probability that a woman has none of the three risk factors, given that she does not have risk factor A. (A) 0.280 (B) 0.311 (C) 0.467 (D) 0.484 (E) 0.700

13. In modeling the number of claims filed by an individual under an automobile policy during a three-year period, an actuary makes the simplifying assumption that for all integers n ≥ 0 , p ( n+1)=0.2p(n) where p(n) represents the probability that the policyholder files n claims during the period.
 Under this assumption, calculate the probability that a policyholder files more than one claim during the period. (A) 0.04 (B) 0.16 (C) 0.20 (D) 0.80 (E) 0.96
14. An insurer offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose exactly two of the supplementary coverages A, B, and C, or they may choose no supplementary coverage. The proportions of the company’s employees that choose coverages A, B, and C are 1/4, 1/3, and 5/12 respectively.
Calculate the probability that a randomly chosen employee will choose no supplementary coverage. (A) 0 (B) 47/144 (C) 1/2 (D) 97/144 (E) 7/9

15. An insurance company determines that N, the number of claims received in a week, is a random variable with P[N=n]= where n ≥ 0 . The company also determines that the number of claims received in a given week is independent of the number of claims received in any other week.
Calculate the probability that exactly seven claims will be received during a given two-week period. (A) 1/256 (B) 1/128 (C) 7/512 (D) 1/64 (E) 1/32
16. An insurance company pays hospital claims. The number of claims that include emergency room or operating room charges is 85% of the total number of claims. The number of claims that do not include emergency room charges is 25% of the total number of claims. The occurrence of emergency room charges is independent of the occurrence of operating room charges on hospital claims.
 Calculate the probability that a claim submitted to the insurance company includes operating room charges. (A) 0.10 (B) 0.20 (C) 0.25 (D) 0.40 (E) 0.80
17. Two instruments are used to measure the height, h, of a tower. The error made by the less accurate instrument is normally distributed with mean 0 and standard deviation 0.0056h. The error made by the more accurate instrument is normally distributed with mean 0 and standard deviation 0.0044h.
 The errors from the two instruments are independent of each other.
Calculate the probability that the average value of the two measurements is within 0.005h of the height of the tower. (A) 0.38 (B) 0.47 (C) 0.68 (D) 0.84 (E) 0.90

18. An auto insurance company insures drivers of all ages. An actuary compiled the following statistics on the company’s insured drivers:

 A randomly selected driver that the company insures has an accident.
Calculate the probability that the driver was age 16-20. (A) 0.13 (B) 0.16 (C) 0.19 (D) 0.23 (E) 0.40
19. An insurance company issues life insurance policies in three separate categories: standard, preferred, and ultra-preferred. Of the company’s policyholders, 50% are standard, 40% are preferred, and 10% are ultra-preferred. Each standard policyholder has probability 0.010 of dying in the next year, each preferred policyholder has probability 0.005 of dying in the next year, and each ultra-preferred policyholder has probability 0.001 of dying in the next year.
A policyholder dies in the next year.
Calculate the probability that the deceased policyholder was ultra-preferred. (A) 0.0001 (B) 0.0010 (C) 0.0071 (D) 0.0141 (E) 0.2817
20. Upon arrival at a hospital’s emergency room, patients are categorized according to their condition as critical, serious, or stable. In the past year:
(i) 10% of the emergency room patients were critical；
(ii) 30% of the emergency room patients were serious；
(iii) the rest of the emergency room patients were stable；
(iv) 40% of the critical patients died；
(vi) 10% of the serious patients died； and
(vii) 1% of the stable patients died.
Given that a patient survived, calculate the probability that the patient was categorized as serious upon arrival.(A) 0.06(B) 0.29(C) 0.30(D) 0.39(E) 0.64
21. A health study tracked a group of persons for five years. At the beginning of the study,20% were classified as heavy smokers, 30% as light smokers, and 50% as nonsmokers.
 Results of the study showed that light smokers were twice as likely as nonsmokers to die during the five-year study, but only half as likely as heavy smokers.
 A randomly selected participant from the study died during the five-year period.
 Calculate the probability that the participant was a heavy smoker. (A) 0.20 (B) 0.25 (C) 0.35 (D) 0.42 (E) 0.57

22. An actuary studied the likelihood that different types of drivers would be involved in at least one collision during any one-year period. The results of the study are:
  
 Given that a driver has been involved in at least one collision in the past year, calculate the probability that the driver is a young adult driver. (A) 0.06 (B) 0.16 (C) 0.19 (D) 0.22 (E) 0.25

23. The number of injury claims per month is modeled by a random variable N with
P[N=n]=, for nonnegative integers, n.
Calculate the probability of at least one claim during a particular month, given that there have been at most four claims during that month. (A) 1/3 (B) 2/5 (C) 1/2 (D) 3/5 (E) 5/6
24. A blood test indicates the presence of a particular disease 95% of the time when the disease is actually present. The same test indicates the presence of the disease 0.5% of the time when the disease is not actually present. One percent of the population actually has the disease.
Calculate the probability that a person actually has the disease given that the test indicates the presence of the disease. (A) 0.324 (B) 0.657 (C) 0.945 (D) 0.950 (E) 0.995
25. The probability that a randomly chosen male has a blood circulation problem is 0.25. Males who have a blood circulation problem are twice as likely to be smokers as those who do not have a blood circulation problem.
Calculate the probability that a male has a blood circulation problem, given that he is a smoker. (A) 1/4 (B) 1/3 (C) 2/5 (D) 1/2 (E) 2/3

26. A study of automobile accidents produced the following data:

 An automobile from one of the model years 2014, 2013, and 2012 was involved in an accident.
Calculate the probability that the model year of this automobile is 2014. (A) 0.22 (B) 0.30 (C) 0.33 (D) 0.45 (E) 0.50
27. A hospital receives 1/5 of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials.
For Company X’s shipments, 10% of the vials are ineffective. For every other company,2% of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective.
Calculate the probability that this shipment came from Company X. (A) 0.10 (B) 0.14 (C) 0.37 (D) 0.63 (E) 0.86
28. The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. An insurance company expects that 30% of high-risk drivers will be involved in an accident during the first 50 days of a calendar year.

Calculate the portion of high-risk drivers are expected to be involved in an accident during the first 80 days of a calendar year. (A) 0.15 (B) 0.34 (C) 0.43 (D) 0.57 (E) 0.66

29. An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims.
The number of claims filed has a Poisson distribution.
Calculate the variance of the number of claims filed. (A)(B) 1 (C)(D) 2 (E) 4
30. A company establishes a fund of 120 from which it wants to pay an amount, C, to any of its 20 employees who achieve a high performance level during the coming year. Each employee has a 2% chance of achieving a high performance level during the coming year. The events of different employees achieving a high performance level during the coming year are mutually independent.

Calculate the maximum value of C for which the probability is less than 1% that the fund will be inadequate to cover all payments for high performance. (A) 24 (B) 30 (C) 40 (D) 60 (E) 120
31. A large pool of adults earning their first driver’s license includes 50% low-risk drivers,30% moderate-risk drivers, and 20% high-risk drivers. Because these drivers have no prior driving record, an insurance company considers each driver to be randomly selected from the pool.

This month, the insurance company writes four new policies for adults earning their first driver’s license.

Calculate the probability that these four will contain at least two more high-risk drivers than low-risk drivers. (A) 0.006 (B) 0.012 (C) 0.018 (D) 0.049 (E) 0.073

32. The loss due to a fire in a commercial building is modeled by a random variable X with density function

   Given that a fire loss exceeds 8, calculate the probability that it exceeds 16. (A) 1/25 (B) 1/9 (C) 1/8 (D) 1/3 (E) 3/7

33. The lifetime of a machine part has a continuous distribution on the interval (0, 40) with probability density function f(x), where f(x) is proportional to on the interval.
Calculate the probability that the lifetime of the machine part is less than 6. (A) 0.04 (B) 0.15 (C) 0.47 (D) 0.53 (E) 0.94

34. A group insurance policy covers the medical claims of the employees of a small company. The value, V, of the claims made in one year is described by
V = 100,000Y
where Y is a random variable with density function

where k is a constant.
Calculate the conditional probability that V exceeds 40,000, given that V exceeds 10,000. (A) 0.08 (B) 0.13 (C) 0.17 (D) 0.20 (E) 0.51
35. The lifetime of a printer costing 200 is exponentially distributed with mean 2 years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, a one-half refund if it fails during the second year, and no refund for failure after the second year.
Calculate the expected total amount of refunds from the sale of 100 printers. (A) 6,321 (B) 7,358 (C) 7,869 (D) 10,256 (E) 12,642

36. An insurance company insures a large number of homes. The insured value, X, of a randomly selected home is assumed to follow a distribution with density function
  
 Given that a randomly selected home is insured for at least 1.5, calculate the probability that it is insured for less than 2. (A) 0.578 (B) 0.684 (C) 0.704 (D) 0.829 (E) 0.875
37. A company prices its hurricane insurance using the following assumptions:
(i) In any calendar year, there can be at most one hurricane.
(ii) In any calendar year, the probability of a hurricane is 0.05.
(iii) The numbers of hurricanes in different calendar years are mutually independent.
Using the company’s assumptions, calculate the probability that there are fewer than 3 hurricanes in a 20-year period.(A) 0.06(B) 0.19(C) 0.38(D) 0.62(E) 0.92

38. An insurance policy pays for a random loss X subject to a deductible of C, where 0＜ C＜ 1 . The loss amount is modeled as a continuous random variable with density function

Given a random loss X, the probability that the insurance payment is less than 0.5 is equal to 0.64. Calculate C. (A) 0.1 (B) 0.3 (C) 0.4 (D) 0.6 (E) 0.8
39. A study is being conducted in which the health of two independent groups of ten policyholders is being monitored over a one-year period of time. Individual participants in the study drop out before the end of the study with probability 0.2 (independently of the other participants).
Calculate the probability that at least nine participants complete the study in one of the two groups, but not in both groups? (A) 0.096 (B) 0.192 (C) 0.235 (D) 0.376 (E) 0.469
40. For Company A there is a 60% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 10,000 and standard deviation 2,000.

For Company B there is a 70% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 9,000 and standard deviation 2,000.

 The total claim amounts of the two companies are independent.

Calculate the probability that, in the coming year, Company B’s total claim amount will exceed Company A’s total claim amount. (A) 0.180 (B) 0.185 (C) 0.217 (D) 0.223 (E) 0.240
41. A company takes out an insurance policy to cover accidents that occur at its manufacturing plant. The probability that one or more accidents will occur during any given month is 0.60. The numbers of accidents that occur in different months are mutually independent.
 Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs. (A) 0.01 (B) 0.12 (C) 0.23 (D) 0.29 (E) 0.41

42. An insurance policy pays 100 per day for up to three days of hospitalization and 50 per day for each day of hospitalization thereafter.
 The number of days of hospitalization, X, is a discrete random variable with probability function

 Determine the expected payment for hospitalization under this policy. (A) 123 (B) 210 (C) 220 (D) 270 (E) 367

43. Let X be a continuous random variable with density function

 Calculate the expected value of X. (A) 1/5 (B) 3/5 (C) 1 (D) 28/15 (E) 12/5

44. A device that continuously measures and records seismic activity is placed in a remote region. The time, T, to failure of this device is exponentially distributed with mean 3 years. Since the device will not be monitored during its first two years of service, the time to discovery of its failure is X = max(T, 2).
Calculate E(X). (A)(B)(C) 3 (D) (E) 5
45. A piece of equipment is being insured against early failure. The time from purchase until failure of the equipment is exponentially distributed with mean 10 years. The insurance will pay an amount x if the equipment fails during the first year, and it will pay 0.5x if failure occurs during the second or third year. If failure occurs after the first three years, no payment will be made.
Calculate x such that the expected payment made under this insurance is 1000. (A) 3858 (B) 4449 (C) 5382 (D) 5644 (E) 7235
46. An insurance policy on an electrical device pays a benefit of 4000 if the device fails during the first year. The amount of the benefit decreases by 1000 each successive year until it reaches 0. If the device has not failed by the beginning of any given year, the probability of failure during that year is 0.4.

Calculate the expected benefit under this policy. (A) 2234 (B) 2400 (C) 2500 (D) 2667 (E) 2694
47. A company buys a policy to insure its revenue in the event of major snowstorms that shut down business. The policy pays nothing for the first such snowstorm of the year and 10,000 for each one thereafter, until the end of the year. The number of major snowstorms per year that shut down business is assumed to have a Poisson distribution with mean 1.5.
Calculate the expected amount paid to the company under this policy during a one-year period. (A) 2,769 (B) 5,000 (C) 7,231 (D) 8,347 (E) 10,578

48. A manufacturer’s annual losses follow a distribution with density function

 To cover its losses, the manufacturer purchases an insurance policy with an annual deductible of 2.
Calculate the mean of the manufacturer’s annual losses not paid by the insurance policy. (A) 0.84 (B) 0.88 (C) 0.93 (D) 0.95 (E) 1.00
49. An insurance company sells a one-year automobile policy with a deductible of 2. The probability that the insured will incur a loss is 0.05. If there is a loss, the probability of a loss of amount N is K/N, for N = 1, . . . , 5 and K a constant. These are the only possible loss amounts and no more than one loss can occur.

Calculate the expected payment for this policy. (A) 0.031 (B) 0.066 (C) 0.072 (D) 0.110 (E) 0.150

50. An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder’s loss, Y, follows a distribution with density function:

  Calculate the expected value of the benefit paid under the insurance policy. (A) 1.0 (B) 1.3 (C) 1.8 (D) 1.9 (E) 2.0

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