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112年 - SOCIETY OF ACTUARIES_EXAM P PROBABILITY_EXAM P SAMPLE QUESTIONS 51-100#119910

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)

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

 Calculate the expected claim payment. (A) 320 (B) 328 (C) 352 (D) 380 (E) 540

52. An insurance company’s monthly claims are modeled by a continuous, positive random variable X, whose probability density function is proportional to  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

Calculate the difference between the 30th and 70th percentiles of X. (A) 35 (B) 93 (C) 124 (D) 231 (E) 298
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

 Calculate the variance of X. (A) 7/72 (B) 1/8 (C) 5/36 (D) 4/3 (E) 23/12

57. The warranty on a machine specifies that it will be replaced at failure or age 4, whichever occurs first. The machine’s age at failure, X, has density function

 Let Y be the age of the machine at the time of replacement.
Calculate the variance of Y. (A) 1.3 (B) 1.4 (C) 1.7 (D) 2.1 (E) 7.5

58. A probability distribution of the claim sizes for an auto insurance policy is given in the table below:

  Calculate the percentage of claims that are within one standard deviation of the mean claim size. (A) 45% (B) 55% (C) 68% (D) 85% (E) 100%
59. The owner of an automobile insures it against damage by purchasing an insurance policy with a deductible of 250. In the event that the automobile is damaged, repair costs can be modeled by a uniform random variable on the interval (0, 1500).
Calculate the standard deviation of the insurance payment in the event that the automobile is damaged. (A) 361 (B) 403 (C) 433 (D) 464 (E) 521
60. A baseball team has scheduled its opening game for April 1. If it rains on April 1, the game is postponed and will be played on the next day that it does not rain. The team purchases insurance against rain. The policy will pay 1000 for each day, up to 2 days, that the opening game is postponed.
The insurance company determines that the number of consecutive days of rain beginning on April 1 is a Poisson random variable with mean 0.6.
Calculate the standard deviation of the amount the insurance company will have to pay. (A) 668 (B) 699 (C) 775 (D) 817 (E) 904

61. An insurance policy reimburses dental expense, X, up to a maximum benefit of 250. The probability density function for X is:

where c is a constant.
Calculate the median benefit for this policy. (A) 161 (B) 165 (C) 173 (D) 182 (E) 250
62. The time to failure of a component in an electronic device has an exponential distribution with a median of four hours.
Calculate the probability that the component will work without failing for at least five hours. (A) 0.07 (B) 0.29 (C) 0.38 (D) 0.42 (E) 0.57

63. An insurance company sells an auto insurance policy that covers losses incurred by a policyholder, subject to a deductible of 100. Losses incurred follow an exponential distribution with mean 300.
Calculate the percentile of losses that exceed the deductible. (A) 600 (B) 700 (C) 800 (D) 900 (E) 1000

64. Claim amounts for wind damage to insured homes are mutually independent random variables with common density function

 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
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
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

 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
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

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)(B)(C)(D)(E)
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
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
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 means1.0, 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(X2)=27.4,E(Y)=7,E(Y2)=51.4,Var(X+Y)=8.
Let C1=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) . (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 is0.025, 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

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:
  
 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:

  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.
 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
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%

88. The cumulative distribution function for health care costs experienced by a policyholder is modeled by the function

 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 at50%.
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 N2 represent the numbers of claims submitted to a life insurance company in April and May, respectively. The joint probability function of N1 and N2 is
 
 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)(B)(C)(D)e2 −1 (E)e2
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

 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
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) 2 λ (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
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.

 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
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 lowrisk 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
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

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