edu-exam-p-sample-quest (12).pdf

SOCIETY OF ACTUARIES EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS This set of sample questions includes those published on the probability topic for use with previous versions of this examination. Qu...

SOCIETY OF ACTUARIES EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS This set of sample questions includes those published on the probability topic for use with previous versions of this examination. Questions from previous versions of this document that are not relevant for the syllabus effective with the September 2022 administration have been deleted. The questions have been renumbered. Unless indicated below, no questions have been added to the version published for use with exams through July 2022. Some of the questions in this study note are taken from past SOA examinations. These questions are representative of the types of questions that might be asked of candidates sitting for the Probability (P) Exam. These questions are intended to represent the depth of understanding required of candidates. The distribution of questions by topic is not intended to represent the distribution of questions on future exams. For questions involving the normal distribution, answer choices have been calculated using exact values for the normal distribution. Using the normal table provided and rounding to the closest value may give you answers slightly different from the answer choices. As always, choose the best answer provided from the five choices when selecting your answer. Questions 271-287 were added July 2022. Questions 288-319 were added August 2022. Questions 234-236 and 282 were deleted October 2022 Questions 320-446 were added November 2023 Several questions that were duplicates of earlier questions were removed February 2024 Questions 447-485 were added March 2024 Copyright 2024 by the Society of Actuaries. Page 1 of 199 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. 0.7 and P[ A ∪ B′] = You are given P[ A ∪ B] = 0.9. Calculate P[A]. (A) 0.2 (B) 0.3 (C) 0.4 (D) 0.6 (E) 0.8 Page 2 of 199 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 Page 3 of 199 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 Page 4 of 199 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 Page 5 of 199 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 Page 6 of 199 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) = 0.2 p ( 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 1 random variable with P[ N= n= ] where n ≥ 0. The company also determines that 2n +1 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 Page 7 of 199 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 Page 8 of 199 18. An auto insurance company insures drivers of all ages. An actuary compiled the following statistics on the company’s insured drivers: Age of Probability Portion of Company’s Driver of Accident Insured Drivers 16-20 0.06 0.08 21-30 0.03 0.15 31-65 0.02 0.49 66-99 0.04 0.28 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 Page 9 of 199 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 Page 10 of 199 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: Probability Type of Percentage of of at least one driver all drivers collision Teen 8% 0.15 Young adult 16% 0.08 Midlife 45% 0.04 Senior 31% 0.05 Total 100% 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 1 P[ N= n=] , for nonnegative integers, n. (n + 1)(n + 2) 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 Page 11 of 199 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 Page 12 of 199 26. A study of automobile accidents produced the following data: Probability of Model Proportion of involvement year all vehicles in an accident 2014 0.16 0.05 2013 0.18 0.02 2012 0.20 0.03 Other 0.46 0.04 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 Page 13 of 199 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. 1 (A) 3 (B) 1 (C) 2 (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 Page 14 of 199 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 0.005(20 − x), 0 < x < 20 f ( x) =  0, otherwise. 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 (10 + x)− 2 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 Page 15 of 199 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 k (1 − y ) 4 , 0 < y < 1 f ( y) =  0, otherwise 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 Page 16 of 199 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 3 x −4 , x > 1 f ( x) =  0, otherwise. 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 Page 17 of 199 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  2 x, 0 < x < 1 f ( x) =  0, otherwise. 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 Page 18 of 199 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 Page 19 of 199 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 6 − k  , k = 1, 2,3, 4,5 P[ X= k=]  15  0, otherwise. 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 | x |  , −2 ≤ x ≤ 4 f ( x) =  10 0, otherwise. Calculate the expected value of X. (A) 1/5 (B) 3/5 (C) 1 (D) 28/15 (E) 12/5 Page 20 of 199 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). 1 (A) 2 + e −6 3 (B) 2 − 2e −2/3 + 5e −4/3 (C) 3 (D) 2 + 3e −2/3 (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 Page 21 of 199 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  2.5(0.6) 2.5  , x > 0.6 f ( x) =  x 3.5 0,  otherwise. 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 Page 22 of 199 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: 2 y −3 , y > 1 f ( y) =  0, otherwise. 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 51. An auto insurance company insures an automobile worth 15,000 for one year under a policy with a 1,000 deductible. During the policy year there is a 0.04 chance of partial damage to the car and a 0.02 chance of a total loss of the car. If there is partial damage to the car, the amount X of damage (in thousands) follows a distribution with density function 0.5003e − x / 2 , 0 < x < 15 f ( x) =  0, otherwise. Calculate the expected claim payment. (A) 320 (B) 328 (C) 352 (D) 380 (E) 540 Page 23 of 199 52. An insurance company’s monthly claims are modeled by a continuous, positive random variable X, whose probability density function is proportional to (1 + x)−4, for 0 < x < ∞. Calculate the company’s expected monthly claims. (A) 1/6 (B) 1/3 (C) 1/2 (D) 1 (E) 3 53. An insurance policy is written to cover a loss, X, where X has a uniform distribution on [0, 1000]. The policy has a deductible, d, and the expected payment under the policy is 25% of what it would be with no deductible. Calculate d. (A) 250 (B) 375 (C) 500 (D) 625 (E) 750 54. An insurer's annual weather-related loss, X, is a random variable with density function  2.5(200) 2.5  , x > 200 f ( x) =  x 3.5 0,  otherwise. Calculate the difference between the 30th and 70th percentiles of X. (A) 35 (B) 93 (C) 124 (D) 231 (E) 298 Page 24 of 199 55. A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 with a variance of 260. A tax of 20% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 20% more expensive). Calculate the variance of the annual cost of maintaining and repairing a car after the tax is introduced. (A) 208 (B) 260 (C) 270 (D) 312 (E) 374 56. A random variable X has the cumulative distribution function 0, x 1 f ( x) =  x 4  0, otherwise, where x is the amount of a claim in thousands. Suppose 3 such claims will be made. Calculate the expected value of the largest of the three claims. (A) 2025 (B) 2700 (C) 3232 (D) 3375 (E) 4500 65. A charity receives 2025 contributions. Contributions are assumed to be mutually independent and identically distributed with mean 3125 and standard deviation 250. Calculate the approximate 90th percentile for the distribution of the total contributions received. (A) 6,328,000 (B) 6,338,000 (C) 6,343,000 (D) 6,784,000 (E) 6,977,000 66. Claims filed under auto insurance policies follow a normal distribution with mean 19,400 and standard deviation 5,000. Calculate the probability that the average of 25 randomly selected claims exceeds 20,000. (A) 0.01 (B) 0.15 (C) 0.27 (D) 0.33 (E) 0.45 Page 29 of 199 67. An insurance company issues 1250 vision care insurance policies. The number of claims filed by a policyholder under a vision care insurance policy during one year is a Poisson random variable with mean 2. Assume the numbers of claims filed by different policyholders are mutually independent. Calculate the approximate probability that there is a total of between 2450 and 2600 claims during a one-year period? (A) 0.68 (B) 0.82 (C) 0.87 (D) 0.95 (E) 1.00 68. A company manufactures a brand of light bulb with a lifetime in months that is normally distributed with mean 3 and variance 1. A consumer buys a number of these bulbs with the intention of replacing them successively as they burn out. The light bulbs have mutually independent lifetimes. Calculate the smallest number of bulbs to be purchased so that the succession of light bulbs produces light for at least 40 months with probability at least 0.9772. (A) 14 (B) 16 (C) 20 (D) 40 (E) 55 Page 30 of 199 69. Let X and Y be the number of hours that a randomly selected person watches movies and sporting events, respectively, during a three-month period. The following information is known about X and Y: E(X) = 50, E(Y) = 20, Var(X) = 50, Var(Y) = 30, Cov(X,Y) = 10. The totals of hours that different individuals watch movies and sporting events during the three months are mutually independent. One hundred people are randomly selected and observed for these three months. Let T be the total number of hours that these one hundred people watch movies or sporting events during this three-month period. Approximate the value of P[T < 7100]. (A) 0.62 (B) 0.84 (C) 0.87 (D) 0.92 (E) 0.97 70. The total claim amount for a health insurance policy follows a distribution with density function 1 −( x /1000) f ( x) = e , x > 0. 1000 The premium for the policy is set at the expected total claim amount plus 100. If 100 policies are sold, calculate the approximate probability that the insurance company will have claims exceeding the premiums collected. (A) 0.001 (B) 0.159 (C) 0.333 (D) 0.407 (E) 0.460 Page 31 of 199 71. A city has just added 100 new female recruits to its police force. The city will provide a pension to each new hire who remains with the force until retirement. In addition, if the new hire is married at the time of her retirement, a second pension will be provided for her husband. A consulting actuary makes the following assumptions: (i) Each new recruit has a 0.4 probability of remaining with the police force until retirement. (ii) Given that a new recruit reaches retirement with the police force, the probability that she is not married at the time of retirement is 0.25. (iii) The events of different new hires reaching retirement and the events of different new hires being married at retirement are all mutually independent events. Calculate the probability that the city will provide at most 90 pensions to the 100 new hires and their husbands. (A) 0.60 (B) 0.67 (C) 0.75 (D) 0.93 (E) 0.99 72. In an analysis of healthcare data, ages have been rounded to the nearest multiple of 5 years. The difference between the true age and the rounded age is assumed to be uniformly distributed on the interval from −2.5 years to 2.5 years. The healthcare data are based on a random sample of 48 people. Calculate the approximate probability that the mean of the rounded ages is within 0.25 years of the mean of the true ages. (A) 0.14 (B) 0.38 (C) 0.57 (D) 0.77 (E) 0.88 Page 32 of 199 73. The waiting time for the first claim from a good driver and the waiting time for the first claim from a bad driver are independent and follow exponential distributions with means 6 years and 3 years, respectively. Calculate the probability that the first claim from a good driver will be filed within 3 years and the first claim from a bad driver will be filed within 2 years. (A) 1 18 (1 − e−2/3 − e−1/ 2 + e−7/6 ) 1 −7/6 (B) e 18 (C) 1 − e −2/3 − e −1/ 2 + e −7/6 (D) 1 − e −2/3 − e −1/ 2 + e −1/3 1 1 1 (E) 1 − e −2/3 − e −1/ 2 + e −7/6 3 6 18 74. A tour operator has a bus that can accommodate 20 tourists. The operator knows that tourists may not show up, so he sells 21 tickets. The probability that an individual tourist will not show up is 0.02, independent of all other tourists. Each ticket costs 50, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay 100 (ticket cost + 50 penalty) to the tourist. Calculate the expected revenue of the tour operator. (A) 955 (B) 962 (C) 967 (D) 976 (E) 985 Page 33 of 199 75. An insurance policy pays a total medical benefit consisting of two parts for each claim. Let X represent the part of the benefit that is paid to the surgeon, and let Y represent the part that is paid to the hospital. The variance of X is 5000, the variance of Y is 10,000, and the variance of the total benefit, X + Y, is 17,000. Due to increasing medical costs, the company that issues the policy decides to increase X by a flat amount of 100 per claim and to increase Y by 10% per claim. Calculate the variance of the total benefit after these revisions have been made. (A) 18,200 (B) 18,800 (C) 19,300 (D) 19,520 (E) 20,670 76. A car dealership sells 0, 1, or 2 luxury cars on any day. When selling a car, the dealer also tries to persuade the customer to buy an extended warranty for the car. Let X denote the number of luxury cars sold in a given day, and let Y denote the number of extended warranties sold. P[X = 0, Y = 0] = 1/6 P[X = 1, Y = 0] = 1/12 P[X = 1, Y = 1] = 1/6 P[X = 2, Y = 0] = 1/12 P[X = 2, Y = 1] = 1/3 P[X = 2, Y = 2] = 1/6 Calculate the variance of X. (A) 0.47 (B) 0.58 (C) 0.83 (D) 1.42 (E) 2.58 Page 34 of 199 77. The profit for a new product is given by Z = 3X – Y – 5. X and Y are independent random variables with Var(X) = 1 and Var(Y) = 2. Calculate Var(Z). (A) 1 (B) 5 (C) 7 (D) 11 (E) 16 78. A company has two electric generators. The time until failure for each generator follows an exponential distribution with mean 10. The company will begin using the second generator immediately after the first one fails. Calculate the variance of the total time that the generators produce electricity. (A) 10 (B) 20 (C) 50 (D) 100 (E) 200 79. In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be mutually independent, exponentially distributed random variables with respective means 1.0, 1.5, and 2.4. Calculate the probability that the maximum of these losses exceeds 3. (A) 0.002 (B) 0.050 (C) 0.159 (D) 0.287 (E) 0.414 80. Let X denote the size of a surgical claim and let Y denote the size of the associated hospital claim. An actuary is using a model in which E= ( X ) 5, E (= X 2 ) 27.4, E= (Y ) 7, E (= Y 2 ) 51.4, Var ( X = + Y ) 8. Let C= 1 X + Y denote the size of the combined claims before the application of a 20% surcharge on the hospital portion of the claim, and let C2 denote the size of the combined claims after the application of that surcharge. Calculate Cov(C1 , C2 ). Page 35 of 199  (A) 8.80 (B) 9.60 (C) 9.76 (D) 11.52 (E) 12.32 81. Two life insurance policies, each with a death benefit of 10,000 and a one-time premium of 500, are sold to a married couple, one for each person. The policies will expire at the end of the tenth year. The probability that only the wife will survive at least ten years is 0.025, the probability that only the husband will survive at least ten years is 0.01, and the probability that both of them will survive at least ten years is 0.96. Calculate the expected excess of premiums over claims, given that the husband survives at least ten years. (A) 350 (B) 385 (C) 397 (D) 870 (E) 897 Page 36 of 199 82. A diagnostic test for the presence of a disease has two possible outcomes: 1 for disease present and 0 for disease not present. Let X denote the disease state (0 or 1) of a patient, and let Y denote the outcome of the diagnostic test. The joint probability function of X and Y is given by: P[X = 0, Y = 0] = 0.800 P[X = 1, Y = 0] = 0.050 P[X = 0, Y = 1] = 0.025 P[X = 1, Y = 1] = 0.125 Calculate Var(Y X = 1). (A) 0.13 (B) 0.15 (C) 0.20 (D) 0.51 (E) 0.71 83. An actuary determines that the annual number of tornadoes in counties P and Q are jointly distributed as follows: Annual number of tornadoes in county Q 0 1 2 3 Annual number 0 0.12 0.06 0.05 0.02 of tornadoes 1 0.13 0.15 0.12 0.03 in county P 2 0.05 0.15 0.10 0.02 Calculate the conditional variance of the annual number of tornadoes in county Q, given that there are no tornadoes in county P. (A) 0.51 (B) 0.84 (C) 0.88 (D) 0.99 (E) 1.76 84. You are given the following information about N, the annual number of claims for a randomly selected insured: 1 1 1 P ( N = 0) = , P ( N = 1) = , P ( N > 1) =. 2 3 6 Let S denote the total annual claim amount for an insured. When N = 1, S is exponentially distributed with mean 5. When N > 1, S is exponentially distributed with mean 8. Page 37 of 199  Calculate P(4 < S < 8). (A) 0.04 (B) 0.08 (C) 0.12 (D) 0.24 (E) 0.25 85. Under an insurance policy, a maximum of five claims may be filed per year by a policyholder. Let p ( n) be the probability that a policyholder files n claims during a given year, where n = 0,1,2,3,4,5. An actuary makes the following observations: i) p ( n) ≥ p ( n + 1) for n = 0, 1, 2, 3, 4. ii) The difference between p ( n) and p (n + 1) is the same for n = 0,1,2,3,4. iii) Exactly 40% of policyholders file fewer than two claims during a given year. Calculate the probability that a random policyholder will file more than three claims during a given year. (A) 0.14 (B) 0.16 (C) 0.27 (D) 0.29 (E) 0.33 Page 38 of 199 86. The amounts of automobile losses reported to an insurance company are mutually independent, and each loss is uniformly distributed between 0 and 20,000. The company covers each such loss subject to a deductible of 5,000. Calculate the probability that the total payout on 200 reported losses is between 1,000,000 and 1,200,000. (A) 0.0803 (B) 0.1051 (C) 0.1799 (D) 0.8201 (E) 0.8575 87. An insurance agent offers his clients auto insurance, homeowners insurance and renters insurance. The purchase of homeowners insurance and the purchase of renters insurance are mutually exclusive. The profile of the agent’s clients is as follows: i) 17% of the clients have none of these three products. ii) 64% of the clients have auto insurance. iii) Twice as many of the clients have homeowners insurance as have renters insurance. iv) 35% of the clients have two of these three products. v) 11% of the clients have homeowners insurance, but not auto insurance. Calculate the percentage of the agent’s clients that have both auto and renters insurance. (A) 7% (B) 10% (C) 16% (D) 25% (E) 28% Page 39 of 199 88. The cumulative distribution function for health care costs experienced by a policyholder is modeled by the function  − x 1 − e 100 , x > 0 F ( x) =   0, otherwise. The policy has a deductible of 20. An insurer reimburses the policyholder for 100% of health care costs between 20 and 120. Health care costs above 120 are reimbursed at 50%. Let G be the cumulative distribution function of reimbursements given that the reimbursement is positive. Calculate G(115). (A) 0.683 (B) 0.727 (C) 0.741 (D) 0.757 (E) 0.777 89. Let N1 and N 2 represent the numbers of claims submitted to a life insurance company in April and May, respectively. The joint probability function of N1 and N 2 is  3  1 n 1−1 − n ( ) n 2 −1  e 1 1 − e− n 1 , n1 =1, 2,3,..., n2 =1, 2,3,... p (n1 , n2 ) =  4  4  0,  otherwise. Calculate the expected number of claims that will be submitted to the company in May, given that exactly 2 claims were submitted in April. (A) 16 ( e − 1) 3 2 3 2 (B) e 16 3e (C) 4−e (D) e2 − 1 (E) e2 Page 40 of 199 90. A store has 80 modems in its inventory, 30 coming from Source A and the remainder from Source B. Of the modems in inventory from Source A, 20% are defective. Of the modems in inventory from Source B, 8% are defective. Calculate the probability that exactly two out of a sample of five modems selected without replacement from the store’s inventory are defective. (A) 0.010 (B) 0.078 (C) 0.102 (D) 0.105 (E) 0.125 91. A man purchases a life insurance policy on his 40th birthday. The policy will pay 5000 if he dies before his 50th birthday and will pay 0 otherwise. The length of lifetime, in years from birth, of a male born the same year as the insured has the cumulative distribution function 0, t≤0  F (t ) =   1 − 1.1t  1 − exp   , t > 0.   1000  Calculate the expected payment under this policy. (A) 333 (B) 348 (C) 421 (D) 549 (E) 574 92. A mattress store sells only king, queen and twin-size mattresses. Sales records at the store indicate that the number of queen-size mattresses sold is one-fourth the number of king and twin-size mattresses combined. Records also indicate that three times as many king-size mattresses are sold as twin-size mattresses. Calculate the probability that the next mattress sold is either king or queen-size. (A) 0.12 (B) 0.15 (C) 0.80 (D) 0.85 (E) 0.95 Page 41 of 199 93. The number of workplace injuries, N, occurring in a factory on any given day is Poisson distributed with mean λ. The parameter λ is a random variable that is determined by the level of activity in the factory, and is uniformly distributed on the interval [0, 3]. Calculate Var(N). (A) λ (B) 2λ (C) 0.75 (D) 1.50 (E) 2.25 94. A fair die is rolled repeatedly. Let X be the number of rolls needed to obtain a 5 and Y the number of rolls needed to obtain a 6. Calculate E(X | Y = 2). (A) 5.0 (B) 5.2 (C) 6.0 (D) 6.6 (E) 6.8 95. A driver and a passenger are in a car accident. Each of them independently has probability 0.3 of being hospitalized. When a hospitalization occurs, the loss is uniformly distributed on [0, 1]. When two hospitalizations occur, the losses are independent. Calculate the expected number of people in the car who are hospitalized, given that the total loss due to hospitalizations from the accident is less than 1. (A) 0.510 (B) 0.534 (C) 0.600 (D) 0.628 (E) 0.800 Page 42 of 199 96. Each time a hurricane arrives, a new home has a 0.4 probability of experiencing damage. The occurrences of damage in different hurricanes are mutually independent. Calculate the mode of the number of hurricanes it takes for the home to experience damage from two hurricanes. (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 97. Thirty items are arranged in a 6-by-5 array as shown. A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23 A24 A25 A26 A27 A28 A29 A30 Calculate the number of ways to form a set of three distinct items such that no two of the selected items are in the same row or same column. (A) 200 (B) 760 (C) 1200 (D) 4560 (E) 7200 Page 43 of 199 98. An auto insurance company is implementing a new bonus system. In each month, if a policyholder does not have an accident, he or she will receive a cash-back bonus of 5 from the insurer. Among the 1,000 policyholders of the auto insurance company, 400 are classified as low- risk drivers and 600 are classified as high-risk drivers. In each month, the probability of zero accidents for high-risk drivers is 0.80 and the probability of zero accidents for low-risk drivers is 0.90. Calculate the expected bonus payment from the insurer to the 1000 policyholders in one year. (A) 48,000 (B) 50,400 (C) 51,000 (D) 54,000 (E) 60,000 99. The probability that a member of a certain class of homeowners with liability and property coverage will file a liability claim is 0.04, and the probability that a member of this class will file a property claim is 0.10. The probability that a member of this class will file a liability claim but not a property claim is 0.01. Calculate the probability that a randomly selected member of this class of homeowners will not file a claim of either type. (A) 0.850 (B) 0.860 (C) 0.864 (D) 0.870 (E) 0.890 Page 44 of 199 100. A survey of 100 TV viewers revealed that over the last year: i) 34 watched CBS. ii) 15 watched NBC. iii) 10 watched ABC. iv) 7 watched CBS and NBC. v) 6 watched CBS and ABC. vi) 5 watched NBC and ABC. vii) 4 watched CBS, NBC, and ABC. viii) 18 watched HGTV, and of these, none watched CBS, NBC, or ABC. Calculate how many of the 100 TV viewers did not watch any of the four channels (CBS, NBC, ABC or HGTV). (A) 1 (B) 37 (C) 45 (D) 55 (E) 82 101. The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of d, the insurance company reduces the expected claim payment by 10%. Calculate the percentage reduction on the variance of the claim payment. (A) 1% (B) 5% (C) 10% (D) 20% (E) 25% Page 45 of 199 102. The number of hurricanes that will hit a certain house in the next ten years is Poisson distributed with mean 4. Each hurricane results in a loss that is exponentially distributed with mean 1000. Losses are mutually independent and independent of the number of hurricanes. Calculate the variance of the total loss due to hurricanes hitting this house in the next ten years. (A) 4,000,000 (B) 4,004,000 (C) 8,000,000 (D) 16,000,000 (E) 20,000,000 103. A motorist makes three driving errors, each independently resulting in an accident with probability 0.25. Each accident results in a loss that is exponentially distributed with mean 0.80. Losses are mutually independent and independent of the number of accidents. The motorist’s insurer reimburses 70% of each loss due to an accident. Calculate the variance of the total unreimbursed loss the motorist experiences due to accidents resulting from these driving errors. (A) 0.0432 (B) 0.0756 (C) 0.1782 (D) 0.2520 (E) 0.4116 104. An automobile insurance company issues a one-year policy with a deductible of 500. The probability is 0.8 that the insured automobile has no accident and 0.0 that the automobile has more than one accident. If there is an accident, the loss before application of the deductible is exponentially distributed with mean 3000. Calculate the 95th percentile of the insurance company payout on this policy. (A) 3466 (B) 3659 (C) 4159 (D) 8487 (E) 8987 Page 46 of 199 105. From 27 pieces of luggage, an airline luggage handler damages a random sample of four. The probability that exactly one of the damaged pieces of luggage is insured is twice the probability that none of the damaged pieces are insured. Calculate the probability that exactly two of the four damaged pieces are insured. (A) 0.06 (B) 0.13 (C) 0.27 (D) 0.30 (E) 0.31 106. Automobile policies are separated into two groups: low-risk and high-risk. Actuary Rahul examines low-risk policies, continuing until a policy with a claim is found and then stopping. Actuary Toby follows the same procedure with high-risk policies. Each low-risk policy has a 10% probability of having a claim. Each high-risk policy has a 20% probability of having a claim. The claim statuses of polices are mutually independent. Calculate the probability that Actuary Rahul examines fewer policies than Actuary Toby. (A) 0.2857 (B) 0.3214 (C) 0.3333 (D) 0.3571 (E) 0.4000 107. Let X represent the number of customers arriving during the morning hours and let Y represent the number of customers arriving during the afternoon hours at a diner. You are given: i) X and Y are Poisson distributed. ii) The first moment of X is less than the first moment of Y by 8. iii) The second moment of X is 60% of the second moment of Y. Calculate the variance of Y. (A) 4 (B) 12 (C) 16 (D) 27 (E) 35 Page 47 of 199 108. In a certain game of chance, a square board with area 1 is colored with sectors of either red or blue. A player, who cannot see the board, must specify a point on the board by giving an x-coordinate and a y-coordinate. The player wins the game if the specified point is in a blue sector. The game can be arranged with any number of red sectors, and the red sectors are designed so that i  9  Ri is the area of the ith red sector. Ri =   , where  20  Calculate the minimum number of red sectors that makes the chance of a player winning less than 20%. (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 109. Automobile claim amounts are modeled by a uniform distribution on the interval [0, 10,000]. Actuary A reports X, the claim amount divided by 1000. Actuary B reports Y, which is X rounded to the nearest integer from 0 to 10. Calculate the absolute value of the difference between the 4th moment of X and the 4th moment of Y. (A) 0 (B) 33 (C) 296 (D) 303 (E) 533 Page 48 of 199 110. The probability of x losses occurring in year 1 is (0.5) x +1 for x = 0,1, 2,. The probability of y losses in year 2 given x losses in year 1 is given by the table: Number of Number of losses in year 2 (y) losses in given x losses in year 1 year 1 (x) 0 1 2 3 4+ 0 0.60 0.25 0.05 0.05 0.05 1 0.45 0.30 0.10 0.10 0.05 2 0.25 0.30 0.20 0.20 0.05 3 0.15 0.20 0.20 0.30 0.15 4+ 0.05 0.15 0.25 0.35 0.20 Calculate the probability of exactly 2 losses in 2 years. (A) 0.025 (B) 0.031 (C) 0.075 (D) 0.100 (E) 0.131 111. Let X be a continuous random variable with density function  p −1  , x >1 f ( x) =  x p  0, otherwise Calculate the value of p such that E(X) = 2. (A) 1 (B) 2.5 (C) 3 (D) 5 (E) There is no such p. Page 49 of 199 112. The figure below shows the cumulative distribution function of a random variable, X. Calculate E(X). (A) 0.00 (B) 0.50 (C) 1.00 (D) 1.25 (E) 2.50 113. Two fair dice are rolled. Let X be the absolute value of the difference between the two numbers on the dice. Calculate the probability that X < 3. (A) 2/9 (B) 1/3 (C) 4/9 (D) 5/9 (E) 2/3 114. An actuary analyzes a company’s annual personal auto claims, M, and annual commercial auto claims, N. The analysis reveals that Var(M) = 1600, Var(N) = 900, and the correlation between M and N is 0.64. Calculate Var(M + N). (A) 768 (B) 2500 (C) 3268 (D) 4036 (E) 4420 Page 50 of 199 115. An auto insurance policy has a deductible of 1 and a maximum claim payment of 5. Auto loss amounts follow an exponential distribution with mean 2. Calculate the expected claim payment made for an auto loss. (A) 0.5e −2 − 0.5e −12 1 − (B) 2e − 7 e − 3 2 1 − (C) 2e − 2e −3 2 1 − 2 (D) 2e 1 − (E) 3e − 2e −3 2 116. A student takes a multiple-choice test with 40 questions. The probability that the student answers a given question correctly is 0.5, independent of all other questions. The probability that the student answers more than N questions correctly is greater than 0.10. The probability that the student answers more than N + 1 questions correctly is less than 0.10. Calculate N using a normal approximation with the continuity correction. (A) 23 (B) 25 (C) 32 (D) 33 (E) 35 117. In each of the months June, July, and August, the number of accidents occurring in that month is modeled by a Poisson random variable with mean 1. In each of the other 9 months of the year, the number of accidents occurring is modeled by a Poisson random variable with mean 0.5. Assume that these 12 random variables are mutually independent. Calculate the probability that exactly two accidents occur in July through November. (A) 0.084 (B) 0.185 (C) 0.251 (D) 0.257 (E) 0.271 Page 51 of 199 118. An airport purchases an insurance policy to offset costs associated with excessive amounts of snowfall. For every full ten inches of snow in excess of 40 inches during the winter season, the insurer pays the airport 300 up to a policy maximum of 700. The following table shows the probability function for the random variable X of annual (winter season) snowfall, in inches, at the airport. Inches [0,20) [20,30) [30,40) [40,50) [50,60) [60,70) [70,80) [80,90) [90,inf) Probability 0.06 0.18 0.26 0.22 0.14 0.06 0.04 0.04 0.00 Calculate the standard deviation of the amount paid under the policy. (A) 134 (B) 235 (C) 271 (D) 313 (E) 352 119. Damages to a car in a crash are modeled by a random variable with density function c( x 2 − 60 x + 800), 0 < x < 20 f ( x) =  0, otherwise where c is a constant. A particular car is insured with a deductible of 2. This car was involved in a crash with resulting damages in excess of the deductible. Calculate the probability that the damages exceeded 10. (A) 0.12 (B) 0.16 (C) 0.20 (D) 0.26 (E) 0.78 Page 52 of 199 120. Two fair dice, one red and one blue, are rolled. Let A be the event that the number rolled on the red die is odd. Let B be the event that the number rolled on the blue die is odd. Let C be the event that the sum of the numbers rolled on the two dice is odd. Determine which of the following is true. (A) A, B, and C are not mutually independent, but each pair is independent. (B) A, B, and C are mutually independent. (C) Exactly one pair of the three events is independent. (D) Exactly two of the three pairs are independent. (E) No pair of the three events is independent. 121. An urn contains four fair dice. Two have faces numbered 1, 2, 3, 4, 5, and 6; one has faces numbered 2, 2, 4, 4, 6, and 6; and one has all six faces numbered 6. One of the dice is randomly selected from the urn and rolled. The same die is rolled a second time. Calculate the probability that a 6 is rolled both times. (A) 0.174 (B) 0.250 (C) 0.292 (D) 0.380 (E) 0.417 122. An insurance agent meets twelve potential customers independently, each of whom is equally likely to purchase an insurance product. Six are interested only in auto insurance, four are interested only in homeowners insurance, and two are interested only in life insurance. The agent makes six sales. Calculate the probability that two are for auto insurance, two are for homeowners insurance, and two are for life insurance. (A) 0.001 (B) 0.024 (C) 0.069 (D) 0.097 (E) 0.500 Page 53 of 199 123. A policyholder has probability 0.7 of having no claims, 0.2 of having exactly one claim, and 0.1 of having exactly two claims. Claim amounts are uniformly distributed on the interval [0, 60] and are independent. The insurer covers 100% of each claim. Calculate the probability that the total benefit paid to the policyholder is 48 or less. (A) 0.320 (B) 0.400 (C) 0.800 (D) 0.892 (E) 0.924 124. In a given region, the number of tornadoes in a one-week period is modeled by a Poisson distribution with mean 2. The numbers of tornadoes in different weeks are mutually independent. Calculate the probability that fewer than four tornadoes occur in a three-week period. (A) 0.13 (B) 0.15 (C) 0.29 (D) 0.43 (E) 0.86 125. An electronic system contains three cooling components that operate independently. The probability of each component’s failure is 0.05. The system will overheat if and only if at least two components fail. Calculate the probability that the system will overheat. (A) 0.007 (B) 0.045 (C) 0.098 (D) 0.135 (E) 0.143 Page 54 of 199 126. An insurance company’s annual profit is normally distributed with mean 100 and variance 400. Let Z be normally distributed with mean 0 and variance 1 and let F be the cumulative distribution function of Z. Determine the probability that the company’s profit in a year is at most 60, given that the profit in the year is positive. (A) 1 – F(2) (B) F(2)/F(5) (C) [1 – F(2)]/F(5) (D) [F(0.25) – F(0.1)]/F(0.25) (E) [F(5) – F(2)]/F(5) 127. In a group of health insurance policyholders, 20% have high blood pressure and 30% have high cholesterol. Of the policyholders with high blood pressure, 25% have high cholesterol. A policyholder is randomly selected from the group. Calculate the probability that a policyholder has high blood pressure, given that the policyholder has high cholesterol. (A) 1/6 (B) 1/5 (C) 1/4 (D) 2/3 (E) 5/6 Page 55 of 199 128. In a group of 25 factory workers, 20 are low-risk and five are high-risk. Two of the 25 factory workers are randomly selected without replacement. Calculate the probability that exactly one of the two selected factory workers is low-risk. (A) 0.160 (B) 0.167 (C) 0.320 (D) 0.333 (E) 0.633 129. The proportion X of yearly dental claims that exceed 200 is a random variable with probability density function 60 x 3 (1 − x) 2 , 0 < x < 1 f ( x) =  0, otherwise. Calculate Var[X/(1 – X)] (A) 149/900 (B) 10/7 (C) 6 (D) 8 (E) 10 130. This year, a medical insurance policyholder has probability 0.70 of having no emergency room visits, 0.85 of having no hospital stays, and 0.61 of having neither emergency room visits nor hospital stays Calculate the probability that the policyholder has at least one emergency room visit and at least one hospital stay this year. (A) 0.045 (B) 0.060 (C) 0.390 (D) 0.667 (E) 0.840 Page 56 of 199 131. An insurer offers a travelers insurance policy. Losses under the policy are uniformly distributed on the interval [0, 5]. The insurer reimburses a policyholder for a loss up to a maximum of 4. Determine the cumulative distribution function, F, of the benefit that the insurer pays a policyholder who experiences exactly one loss under the policy. 0, x 0. ( x − 1)( x − 2)(12 − x) (A) 990 ( x − 1)( x − 2)(12 − x) (B) 495 ( x − 1)(12 − x)(11 − x) (C) 495 ( x − 1)(12 − x)(11 − x) (D) 990 (10 − x)(12 − x)(11 − x) (E) 990 180. An insurance policy covers losses incurred by a policyholder, subject to a deductible of 10,000. Incurred losses follow a normal distribution with mean 12,000 and standard deviation c. The probability that a loss is less than k is 0.9582, where k is a constant. Given that the loss exceeds the deductible, there is a probability of 0.9500 that it is less than k. Calculate c. (A) 2045 (B) 2267 (C) 2393 (D) 2505 (E) 2840 Page 77 of 199 181. Losses covered by an insurance policy are modeled by a uniform distribution on the interval [0, 1000]. An insurance company reimburses losses in excess of a deductible of 250. Calculate the difference between the median and the 20th percentile of the insurance company reimbursement, over all losses. (A) 225 (B) 250 (C) 300 (D) 375 (E) 500 182. An insurance agent’s files reveal the following facts about his policyholders: i) 243 own auto insurance. ii) 207 own homeowner insurance. iii) 55 own life insurance and homeowner insurance. iv) 96 own auto insurance and homeowner insurance. v) 32 own life insurance, auto insurance and homeowner insurance. vi) 76 more clients own only auto insurance than only life insurance. vii) 270 own only one of these three insurance products. Calculate the total number of the agent’s policyholders who own at least one of these three insurance products. (A) 389 (B) 407 (C) 423 (D) 448 (E) 483 Page 78 of 199 183. A profile of the investments owned by an agent’s clients follows: i) 228 own annuities. ii) 220 own mutual funds. iii) 98 own life insurance and mutual funds. iv) 93 own annuities and mutual funds. v) 16 own annuities, mutual funds, and life insurance. vi) 45 more clients own only life insurance than own only annuities. vii) 290 own only one type of investment (i.e., annuity, mutual fund, or life insurance). Calculate the agent’s total number of clients. (A) 455 (B) 495 (C) 496 (D) 500 (E) 516 184. An actuary compiles the following information from a portfolio of 1000 homeowners insurance policies: i) 130 policies insure three-bedroom homes. ii) 280 policies insure one-story homes. iii) 150 policies insure two-bath homes. iv) 30 policies insure three-bedroom, two-bath homes. v) 50 policies insure one-story, two-bath homes. vi) 40 policies insure three-bedroom, one-story homes. vii) 10 policies insure three-bedroom, one-story, two-bath homes. Calculate the number of homeowners policies in the portfolio that insure neither one- story nor two-bath nor three-bedroom homes. (A) 310 (B) 450 (C) 530 (D) 550 (E) 570 Page 79 of 199 185. Each week, a subcommittee of four individuals is formed from among the members of a committee comprising seven individuals. Two subcommittee members are then assigned to lead the subcommittee, one as chair and the other as secretary. Calculate the maximum number of consecutive weeks that can elapse without having the subcommittee contain four individuals who have previously served together with the same subcommittee chair. (A) 70 (B) 140 (C) 210 (D) 420 (E) 840 186. Bowl I contains eight red balls and six blue balls. Bowl II is empty. Four balls are selected at random, without replacement, and transferred from bowl I to bowl II. One ball is then selected at random from bowl II. Calculate the conditional probability that two red balls and two blue balls were transferred from bowl I to bowl II, given that the ball selected from bowl II is blue. (A) 0.21 (B) 0.24 (C) 0.43 (D) 0.49 (E) 0.57 Page 80 of 199 187. An actuary has done an analysis of all policies that cover two cars. 70% of the policies are of type A for both cars, and 30% of the policies are of type B for both cars. The number of claims on different cars across all policies are mutually independent. The distributions of the number of claims on a car are given in the following table. Number of Type A Type B Claims 0 40% 25% 1 30% 25% 2 20% 25% 3 10% 25% Four policies are selected at random. Calculate the probability that exactly one of the four policies has the same number of claims on both covered cars. (A) 0.104 (B) 0.250 (C) 0.285 (D) 0.417 (E) 0.739 188. A company sells two types of life insurance policies (P and Q) and one type of health insurance policy. A survey of potential customers revealed the following: i) No survey participant wanted to purchase both life policies. ii) Twice as many survey participants wanted to purchase life policy P as life policy Q. iii) 45% of survey participants wanted to purchase the health policy. iv) 18% of survey participants wanted to purchase only the health policy. v) The event that a survey participant wanted to purchase the health policy was independent of the event that a survey participant wanted to purchase a life policy. Calculate the probability that a randomly selected survey participant wanted to purchase exactly one policy. (A) 0.51 (B) 0.60 (C) 0.69 (D) 0.73 (E) 0.78 Page 81 of 199 189. A state is starting a lottery game. To enter this lottery, a player uses a machine that randomly selects six distinct numbers from among the first 30 positive integers. The lottery randomly selects six distinct numbers from the same 30 positive integers. A winning entry must match the same set of six numbers that the lottery selected. The entry fee is 1, each winning entry receives a prize amount of 500,000, and all other entries receive no prize. Calculate the probability that the state will lose money, given that 800,000 entries are purchased. (A) 0.33 (B) 0.39 (C) 0.61 (D) 0.67 (E) 0.74 190. A life insurance company has found there is a 3% probability that a randomly selected application contains an error. Assume applications are mutually independent in this respect. An auditor randomly selects 100 applications. Calculate the probability that 95% or less of the selected applications are error-free. (A) 0.08 (B) 0.10 (C) 0.13 (D) 0.15 (E) 0.18 Page 82 of 199 191. Let A, B, and C be events such that P[A] = 0.2, P[B] = 0.1, and P[C] = 0.3. The events A and B are independent, the events B and C are independent, and the events A and C are mutually exclusive. Calculate P[A ∪ B ∪ C]. (A) 0.496 (B) 0.540 (C) 0.544 (D) 0.550 (E) 0.600 192. The annual numbers of thefts a homeowners insurance policyholder experiences are analyzed over three years. Define the following events: i) A = the event that the policyholder experiences no thefts in the three years. ii) B = the event that the policyholder experiences at least one theft in the second year. iii) C = the event that the policyholder experiences exactly one theft in the first year. iv) D = the event that the policyholder experiences no thefts in the third year. v) E = the event that the policyholder experiences no thefts in the second year, and at least one theft in the third year. Determine which three events satisfy the condition that the probability of their union equals the sum of their probabilities. (A) Events A, B, and E (B) Events A, C, and E (C) Events A, D, and E (D) Events B, C, and D (E) Events B, C, and E Page 83 of 199 193. Four letters to different insureds are prepared along with accompanying envelopes. The letters are put into the envelopes randomly. Calculate the probability that at least one letter ends up in its accompanying envelope. (A) 27/256 (B) 1/4 (C) 11/24 (D) 5/8 (E) 3/4 194. A health insurance policy covers visits to a doctor’s office. Each visit costs 100. The annual deductible on the policy is 350. For a policy, the number of visits per year has the following probability distribution: Number of Visits 0 1 2 3 4 5 6 Probability 0.60 0.15 0.10 0.08 0.04 0.02 0.01 A policy is selected at random from those where costs exceed the deductible. Calculate the probability that this policyholder had exactly five office visits. (A) 0.050 (B) 0.133 (C) 0.286 (D) 0.333 (E) 0.429 195. A machine has two parts labelled A and B. The probability that part A works for one year is 0.8 and the probability that part B works for one year is 0.6. The probability that at least one part works for one year is 0.9. Calculate the probability that part B works for one year, given that part A works for one year. (A) 1/2 (B) 3/5 (C) 5/8 (D) 3/4 (E) 5/6 Page 84 of 199 196. Six claims are to be randomly selected from a group of thirteen different claims, which includes two workers compensation claims, four homeowners claims and seven auto claims. Calculate the probability that the six claims selected will include one workers compensation claim, two homeowners claims and three auto claims. (A) 0.025 (B) 0.107 (C) 0.153 (D) 0.245 (E) 0.643 197. A drawer contains four pairs of socks, with each pair a different color. One sock at a time is randomly drawn from the drawer until a matching pair is obtained. Calculate the probability that the maximum number of draws is required. (A) 0.0006 (B) 0.0095 (C) 0.0417 (D) 0.1429 (E) 0.2286 198. At a mortgage company, 60% of calls are answered by an attendant. The remaining 40% of callers leave their phone numbers. Of these 40%, 75% receive a return phone call the same day. The remaining 25% receive a return call the next day. Of those who initially spoke to an attendant, 80% will apply for a mortgage. Of those who received a return call the same day, 60% will apply. Of those who received a return call the next day, 40% will apply. Calculate the probability that a person initially spoke to an attendant, given that he or she applied for a mortgage. (A) 0.06 (B) 0.26 (C) 0.48 (D) 0.60 (E) 0.69 Page 85 of 199 199. An insurance company studies back injury claims from a manufacturing company. The insurance company finds that 40% of workers do no lifting on the job, 50% do moderate lifting and 10% do heavy lifting. During a given year, the probability of filing a claim is 0.05 for a worker who does no lifting, 0.08 for a worker who does moderate lifting and 0.20 for a worker who does heavy lifting. A worker is chosen randomly from among those who have filed a back injury claim. Calculate the probability that the worker’s job involves moderate or heavy lifting. (A) 0.75 (B) 0.81 (C) 0.85 (D) 0.86 (E) 0.89 200. The number of traffic accidents occurring on any given day in Coralville is Poisson distributed with mean 5. The probability that any such accident involves an uninsured driver is 0.25, independent of all other such accidents. Calculate the probability that on a given day in Coralville there are no traffic accidents that involve an uninsured driver. (A) 0.007 (B) 0.010 (C) 0.124 (D) 0.237 (E) 0.287 Page 86 of 199 201. A group of 100 patients is tested, one patient at a time, for three risk factors for a certain disease until either all patients have been tested or a patient tests positive for more than one of these three risk factors. For each risk factor, a patient tests positive with probability p, where 0 < p < 1. The outcomes of the tests across all patients and all risk factors are mutually independent. Determine an expression for the probability that exactly n patients are tested, where n is a positive integer less than 100. n −1 (A) 1 − 3 p 2 (1 − p)  3 p 2 (1 − p)  n −1 (B) 1 − 3 p 2 (1 − p) − p 3  3 p 2 (1 − p) + p 3  n −1 (C) 1 − 3 p 2 (1 − p) − p 3  3 p 2 (1 − p) + p 3  n −1 (D) n 1 − 3 p 2 (1 − p) − p 3  3 p 2 (1 − p) + p 3  2 3 (E) 3 (1 − p)n −1 p  1 − (1 − p)n −1 p  + (1 − p)n −1 p  202. A representative of a market research firm contacts consumers by phone in order to conduct surveys. The specific consumer contacted by each phone call is randomly determined. The probability that a phone call produces a completed survey is 0.25. Calculate the probability that more than three phone calls are required to produce one completed survey. (A) 0.32 (B) 0.42 (C) 0.44 (D) 0.56 (E) 0.58 Page 87 of 199 203. Four distinct integers are chosen randomly and without replacement from the first twelve positive integers. X is the random variable representing the second smallest of the four selected integers, and p is the probability function of X. Determine p(x) for x = 2,3,…,10. ( x − 1)(11 − x)(12 − x) (A) 495 ( x − 1)(11 − x)(12 − x) (B) 990 ( x − 1)( x − 2)(12 − x) (C) 990 ( x − 1)( x − 2)(12 − x) (D) 495 (10 − x)(11 − x)(12 − x) (E) 495 204. Losses due to burglary are exponentially distributed with mean 100. The probability that a loss is between 40 and 50 equals the probability that a loss is between 60 and r, with r > 60. Calculate r. (A) 68.26 (B) 70.00 (C) 70.51 (D) 72.36 (E) 75.00 Page 88 of 199 205. The time until the next car accident for a particular driver is exponentially distributed with a mean of 200 days. Calculate the probability that the driver has no accidents in the next 365 days, but then has at least one accident in the 365-day period that follows this initial 365-day period. (A) 0.026 (B) 0.135 (C) 0.161 (D) 0.704 (E) 0.839 206. The annual profit of a life insurance company is normally distributed. The probability that the annual profit does not exceed 2000 is 0.7642. The probability that the annual profit does not exceed 3000 is 0.9066. Calculate the probability that the annual profit does not exceed 1000. (A) 0.1424 (B) 0.3022 (C) 0.5478 (D) 0.6218 (E) 0.7257 Page 89 of 199 207. Individuals purchase both collision and liability insurance on their automobiles. The value of the insured’s automobile is V. Assume the loss L on an automobile claim is a random variable with cumulative distribution function  3  l 3    , 0 ≤ l 0 f ( x) =  0, otherwise. Calculate P[ X ≤ 0.5 X ≤ 1.0]. (A) 0.433 (B) 0.547 (C) 0.632 (D) 0.731 (E) 0.865 Page 90 of 199 210. Events E and F are independent. P[E] = 0.84 and P[F] = 0.65. Calculate the probability that exactly one of the two events occurs. (A) 0.056 (B) 0.398 (C) 0.546 (D) 0.650 (E) 0.944 211. A flood insurance company determines that N, the number of claims received in a month, 2 is a random variable with P[ N= n= ] n +1 = 0,1, 2,. The numbers of claims , for n 3 received in different months are mutually independent. Calculate the probability that more than three claims will be received during a consecutive two-month period, given that fewer than two claims were received in the first of the two months. (A) 0.0062 (B) 0.0123 (C) 0.0139 (D) 0.0165 (E) 0.0185 212. Patients in a study are tested for sleep apnea, one at a time, until a patient is found to have this disease. Each patient independently has the same probability of having sleep apnea. Let r represent the probability that at least four patients are tested. Determine the probability that at least twelve patients are tested given that at least four patients are tested. 11 (A) r3 (B) r3 8 3 (C) r (D) r2 1 (E) r3 Page 91 of 199 213. A factory tests 100 light bulbs for defects. The probability that a bulb is defective is 0.02. The occurrences of defects among the light bulbs are mutually independent events. Calculate the probability that exactly two are defective given that the number of defective bulbs is two or fewer. (A) 0.133 (B) 0.271 (C) 0.273 (D) 0.404 (E) 0.677 214. A certain town experiences an average of 5 tornadoes in any four year period. The number of years from now until the town experiences its next tornado as well as the number of years between tornados have identical exponential distributions and all such times are mutually independent Calculate the median number of years from now until the town experiences its next tornado. (A) 0.55 (B) 0.73 (C) 0.80 (D) 0.87 (E) 1.25 215. Losses under an insurance policy are exponentially distributed with mean 4. The deductible is 1 for each loss. Calculate the median amount that the insurer pays a policyholder for a loss under the policy. (A) 1.77 (B) 2.08 (C) 2.12 (D) 2.77 (E) 3.12 Page 92 of 199 216. A company has purchased a policy that will compensate for the loss of revenue due to severe weather events. The policy pays 1000 for each severe weather event in a year after the first two such events in that year. The number of severe weather events per year has a Poisson distribution with mean 1. Calculate the expected amount paid to this company in one year. (A) 80 (B) 104 (C) 368 (D) 512 (E) 632 217. A company provides each of its employees with a death benefit of 100. The company purchases insurance that pays the cost of total death benefits in excess of 400 per year. The number of employees who will die during the year is a Poisson random variable with mean 2. Calculate the expected annual cost to the company of providing the death benefits, excluding the cost of the insurance. (A) 171 (B) 189 (C) 192 (D) 200 (E) 208 218. The number of burglaries occurring on Burlington Street during a one-year period is Poisson distributed with mean 1. Calculate the expected number of burglaries on Burlington Street in a one-year period, given that there are at least two burglaries. (A) 0.63 (B) 2.39 (C) 2.54 (D) 3.00 (E) 3.78 Page 93 of 199 219. For a certain health insurance policy, losses are uniformly distributed on the interval [0, 450]. The policy has a deductible of d and the expected value of the unreimbursed portion of a loss is 56. Calculate d. (A) 60 (B) 87 (C) 112 (D) 169 (E) 224 220. A motorist just had an accident. The accident is minor with probability 0.75 and is otherwise major. Let b be a positive constant. If the accident is minor, then the loss amount follows a uniform distribution on the interval [0, b]. If the accident is major, then the loss amount follows a uniform distribution on the interval [b, 3b]. The median loss amount due to this accident is 672. Calculate the mean loss amount due to this accident.