112年 - SOCIETY OF ACTUARIES_EXAM P PROBABILITY_EXAM P SAMPLE QUESTIONS 151-200#119924

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)

151. The following information is given about a group of high-risk borrowers.
i) Of all these borrowers, 30% defaulted on at least one student loan.
ii) Of the borrowers who defaulted on at least one car loan, 40% defaulted on at least one student loan.
iii) Of the borrowers who did not default on any student loans, 28% defaulted on at least one car loan.
A statistician randomly selects a borrower from this group and observes that the selected borrower defaulted on at least one student loan.
Calculate the probability that the selected borrower defaulted on at least one car loan.(A) 0.33(B) 0.40(C) 0.44(D) 0.65(E) 0.72
152. An insurance company issues policies covering damage to automobiles. The amount of damage is modeled by a uniform distribution on [0, b].

The policy payout is subject to a deductible of b/10.

A policyholder experiences automobile damage.

Calculate the ratio of the standard deviation of the policy payout to the standard deviation of the amount of the damage. (A) 0.8100 (B) 0.9000 (C) 0.9477 (D) 0.9487 (E) 0.9735
153. A policyholder purchases automobile insurance for two years. Define the following events:
 F = the policyholder has exactly one accident in year one.
 G = the policyholder has one or more accidents in year two.
Define the following events:
i) The policyholder has exactly one accident in year one and has more than one accident in year two.
ii) The policyholder has at least two accidents during the two-year period.
iii) The policyholder has exactly one accident in year one and has at least one accident in year two.
iv) The policyholder has exactly one accident in year one and has a total of two or more accidents in the two-year period.
v) The policyholder has exactly one accident in year one and has more accidents in year two than in year one. Determine the number of events from the above list of five that are the same as F ∩ G. (A) None (B) Exactly one (C) Exactly two (D) Exactly three (E) All
154. An insurance company categorizes its policyholders into three mutually exclusive groups: high-risk, medium-risk, and low-risk. An internal study of the company showed that 45% of the policyholders are low-risk and 35% are medium-risk. The probability of death over the next year, given that a policyholder is high-risk is two times the probability of death of a medium-risk policyholder. The probability of death over the next year, given that a policyholder is medium-risk is three times the probability of death of a low-risk policyholder. The probability of death of a randomly selected policyholder, over the next year, is 0.009.

 Calculate the probability of death of a policyholder over the next year, given that the policyholder is high-risk. (A) 0.0025 (B) 0.0200 (C) 0.1215 (D) 0.2000 (E) 0.3750
155. A policy covers a gas furnace for one year. During that year, only one of three problems can occur:
i) The igniter switch may need to be replaced at a cost of 60. There is a 0.10 probability of this.
ii) The pilot light may need to be replaced at a cost of 200. There is a 0.05 probability of this.
iii) The furnace may need to be replaced at a cost of 3000. There is a 0.01 probability of this.

Calculate the deductible that would produce an expected claim payment of 30. (A) 100 (B) At least 100 but less than 150 (C) At least 150 but less than 200 (D) At least 200 but less than 250 (E) At least 250
156. On a block of ten houses, k are not insured. A tornado randomly damages three houses on the block.

The probability that none of the damaged houses are insured is 1/120.

Calculate the probability that at most one of the damaged houses is insured. (A) 1/5 (B) 7/40 (C) 11/60 (D) 49/60 (E) 119/120
157. In a casino game, a gambler selects four different numbers from the first twelve positive integers. The casino then randomly draws nine numbers without replacement from the first twelve positive integers. The gambler wins the jackpot if the casino draws all four of the gambler’s selected numbers.

Calculate the probability that the gambler wins the jackpot. (A) 0.002 (B) 0.255 (C) 0.296 (D) 0.573 (E) 0.625
158. The number of days an employee is sick each month is modeled by a Poisson distribution with mean 1. The numbers of sick days in different months are mutually independent.

Calculate the probability that an employee is sick more than two days in a three-month period. (A) 0.199 (B) 0.224 (C) 0.423 (D) 0.577 (E) 0.801
159. The number of traffic accidents per week at intersection Q has a Poisson distribution with mean 3. The number of traffic accidents per week at intersection R has a Poisson distribution with mean 1.5.

Let A be the probability that the number of accidents at intersection Q exceeds its mean.
Let B be the corresponding probability for intersection R.

Calculate B – A. (A) 0.00 (B) 0.09 (C) 0.13 (D) 0.19 (E) 0.31
160. Losses due to accidents at an amusement park are exponentially distributed. An insurance company offers the park owner two different policies, with different premiums, to insure against losses due to accidents at the park.

Policy A has a deductible of 1.44. For a random loss, the probability is 0.640 that under this policy, the insurer will pay some money to the park owner. Policy B has a deductible of d. For a random loss, the probability is 0.512 that under this policy, the insurer will pay some money to the park owner.

Calculate d. (A) 0.960 (B) 1.152 (C) 1.728 (D) 1.800 (E) 2.160

161. The distribution of the size of claims paid under an insurance policy has probability density function

 Where a ＞0 and c ＞ 0.

For a randomly selected claim, the probability that the size of the claim is less than 3.75 is 0.4871.

Calculate the probability that the size of a randomly selected claim is greater than 4. (A) 0.404 (B) 0.428 (C) 0.500 (D) 0.572 (E) 0.596
162. Company XYZ provides a warranty on a product that it produces. Each year, the number of warranty claims follows a Poisson distribution with mean c. The probability that no warranty claims are received in any given year is 0.60.

Company XYZ purchases an insurance policy that will reduce its overall warranty claim payment costs. The insurance policy will pay nothing for the first warranty claim received and 5000 for each claim thereafter until the end of the year.

Calculate the expected amount of annual insurance policy payments to Company XYZ. (A) 554 (B) 872 (C) 1022 (D) 1354 (E) 1612
163. For a certain health insurance policy, losses are uniformly distributed on the interval [0, b]. The policy has a deductible of 180 and the expected value of the unreimbursed portion of a loss is 144.

Calculate b. (A) 236 (B) 288 (C) 388 (D) 450 (E) 468
164. The working lifetime, in years, of a particular model of bread maker is normally distributed with mean 10 and variance 4.

Calculate the 12th percentile of the working lifetime, in years. (A) 5.30 (B) 7.65 (C) 8.41 (D) 12.35 (E) 14.70
165. The profits of life insurance companies A and B are normally distributed with the same mean. The variance of company B's profit is 2.25 times the variance of company A's profit. The 14th percentile of company A’s profit is the same as the pth percentile of company B’s profit.

Calculate p. (A) 5.3 (B) 9.3 (C) 21.0 (D) 23.6 (E) 31.6

166. The distribution of values of the retirement package offered by a company to new employees is modeled by the probability density function 

 Calculate the variance of the retirement package value for a new employee, given that the value is at least 10. (A) 15 (B) 20 (C) 25 (D) 30 (E) 35
167. Insurance companies A and B each earn an annual profit that is normally distributed with the same positive mean.The standard deviation of company A’s annual profit is one half of its mean.

In a given year, the probability that company B has a loss (negative profit) is 0.9 times the probability that company A has a loss.

Calculate the ratio of the standard deviation of company B’s annual profit to the standard deviation of company A’s annual profit. (A) 0.49 (B) 0.90 (C) 0.98 (D) 1.11 (E) 1.71
168. Claim amounts at an insurance company are independent of one another. In year one, claim amounts are modeled by a normal random variable X with mean 100 and standard deviation 25. In year two, claim amounts are modeled by the random variable Y = 1.04X + 5.

Calculate the probability that a random sample of 25 claim amounts in year two average between 100 and 110. (A) 0.48 (B) 0.53 (C) 0.54 (D) 0.67 (E) 0.68

169. An insurance company will cover losses incurred from tornadoes in a single calendar year. However, the insurer will only cover losses for a maximum of three separate tornadoes during this timeframe. Let X be the number of tornadoes that result in at least 50 million in losses, and let Y be the total number of tornadoes. The joint probability function for X and Y is
  
 where c is a constant.
Calculate the expected number of tornadoes that result in fewer than 50 million in losses. (A) 0.19 (B) 0.28 (C) 0.76 (D) 1.00 (E) 1.10
170. At a polling booth, ballots are cast by ten voters, of whom three are Republicans, two are Democrats, and five are Independents. A local journalist interviews two of these voters, chosen randomly.

Calculate the expectation of the absolute value of the difference between the number of Republicans interviewed and the number of Democrats interviewed. (A) 1/5 (B) 7/15 (C) 3/5 (D) 11/15 (E) 1
171. The random variables X and Y have joint probability function p(x,y) for x = 0,1 and y = 0,1,2.

Suppose 3p(1,1) = p(1,2), and p(1,1) maximizes the variance of XY.

Calculate the probability that X or Y is 0. (A) 11/25 (B) 23/50 (C) 23/49 (D) 26/49 (E) 14/25
172. The number of severe storms that strike city J in a year follows a binomial distribution with n = 5 and p = 0.6. Given that m severe storms strike city J in a year, the number of severe storms that strike city K in the same year is m with probability 1/2, m +1 with probability 1/3, and m + 2 with probability 1/6.

Calculate the expected number of severe storms that strike city J in a year during which 5 severe storms strike city K. (A) 3.5 (B) 3.7 (C) 3.9 (D) 4.0 (E) 5.7

173. Let N denote the number of accidents occurring during one month on the northbound side of a highway and let S denote the number occurring on the southbound side.

Suppose that N and S are jointly distributed as indicated in the table.
  
Calculate Var (N | N + S = 2). (A) 0.48 (B) 0.55 (C) 0.67 (D) 0.91 (E) 1.25

174. An insurance company has an equal number of claims in each of three territories. In each territory, only three claim amounts are possible: 100, 500, and 1000. Based on the company’s data, the probabilities of each claim amount are:

  Calculate the standard deviation of a randomly selected claim amount. (A) 254 (B) 291 (C) 332 (D) 368 (E) 396
175. At the start of a week, a coal mine has a high-capacity storage bin that is half full. During the week, 20 loads of coal are added to the storage bin. Each load of coal has a volume that is normally distributed with mean 1.50 cubic yards and standard deviation 0.25 cubic yards.

During the same week, coal is removed from the storage bin and loaded into 4 railroad cars.

The amount of coal loaded into each railroad car is normally distributed with mean 7.25 cubic yards and standard deviation 0.50 cubic yards. The amounts added to the storage bin or removed from the storage bin are mutually independent.

Calculate the probability that the storage bin contains more coal at the end of the week than it had at the beginning of the week. (A) 0.56 (B) 0.63 (C) 0.67 (D) 0.75 (E) 0.98

176. An insurance company insures a good driver and a bad driver on the same policy. The table below gives the probability of a given number of claims occurring for each of these drivers in the next ten years.

  The number of claims occurring for the two drivers are independent.

Calculate the mode of the distribution of the total number of claims occurring on this policy in the next ten years. (A) 0 (B) 1 (C) 2 (D) 3 (E) 4
177. In a group of 15 health insurance policyholders diagnosed with cancer, each policyholder has probability 0.90 of receiving radiation and probability 0.40 of receiving chemotherapy. Radiation and chemotherapy treatments are independent events for each policyholder, and the treatments of different policyholders are mutually independent.

The policyholders in this group all have the same health insurance that pays 2 for radiation treatment and 3 for chemotherapy treatment.

Calculate the variance of the total amount the insurance company pays for the radiation and chemotherapy treatments for these 15 policyholders. (A) 13.5 (B) 37.8 (C) 108.0 (D) 202.5 (E) 567.0
178. In a large population of patients, 20% have early stage cancer, 10% have advanced stage cancer, and the other 70% do not have cancer. Six patients from this population are randomly selected.

Calculate the expected number of selected patients with advanced stage cancer, given that at least one of the selected patients has early stage cancer. (A) 0.403 (B) 0.500 (C) 0.547 (D) 0.600 (E) 0.625

179. Four distinct integers are chosen randomly and without replacement from the first twelve positive integers. Let X be the random variable representing the second largest of the four selected integers, and let p be the probability function for X.
Determine p(x), for integer values of x, where p(x) ＞ 0. (A)(B)(C)(D)(E)
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
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
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 onestory nor two-bath nor three-bedroom homes.(A) 310(B) 450(C) 530(D) 550(E) 570
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

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.

 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
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
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
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:
  
 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
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
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

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