112年 - SOCIETY OF ACTUARIES_EXAM P PROBABILITY_EXAM P SAMPLE QUESTIONS 251-319#119927

251. Two independent estimates are to be made on a building damaged by fire. Each estimate is normally distributed with mean 10b and variance b² .

Calculate the probability that the first estimate is at least 20 percent higher than the second. (A) 0.023 (B) 0.100 (C) 0.115 (D) 0.221 (E) 0.444

252. The independent random variables X and Y have the same mean. The coefficients of variation of X and Y are 3 and 4 respectively.

Calculate the coefficient of variation of . (A) 5/4 (B) 7/4 (C) 5/2 (D) 7/2 (E) 7

253. Points scored by a game participant can be modeled by Z = 3X + 2Y – 5. X and Y are independent random variables with Var (X) = 3 and Var (Y) = 4.

Calculate Var (Z). (A) 12 (B) 17 (C) 38 (D) 43 (E) 68

254. An actuary is studying hurricane models. A year is classified as a high, medium, or low hurricane year with probabilities 0.1, 0.3, and 0.6, respectively. The numbers of hurricanes in high, medium, and low years follow Poisson distributions with means 20, 15, and 10, respectively.

Calculate the variance of the number of hurricanes in a randomly selected year. (A) 11.25 (B) 12.50 (C) 12.94 (D) 13.42 (E) 23.75

255. A dental insurance company pays 100% of the cost of fillings and 70% of the cost of root canals. Fillings and root canals cost 50 and 500 each, respectively.

The tables below show the probability distributions of the annual number of fillings and annual number of root canals for each of the company’s policyholders.

Calculate the expected annual payment per policyholder for fillings and root canals. (A) 90.00 (B) 102.50 (C) 132.50 (D) 250.00 (E) 400.00

256. A loss under a liability policy is modeled by an exponential distribution. The insurance company will cover the amount of that loss in excess of a deductible of 2000. The probability that the reimbursement is less than 6000, given that the loss exceeds the deductible, is 0.50.

Calculate the probability that the reimbursement is greater than 3000 but less than 9000, given that the loss exceeds the deductible. (A) 0.28 (B) 0.35 (C) 0.50 (D) 0.65 (E) 0.72

257. Let X be the percentage score on a college-entrance exam for students who did not participate in an exam-preparation seminar. X is modeled by a uniform distribution on [a, 100]. Let Y be the percentage score on a college-entrance exam for students who did participate in an exam-preparation seminar. Y is modeled by a uniform distribution on [1.25a, 100]. It is given that .
Calculate the percentile of Y. (A) 80 (B) 85 (C) 90 (D) 92 (E) 95

258. In a study of driver safety, drivers were categorized according to three risk factors. Exactly 1000 drivers exhibited each individual risk factor. Also, for each of the risk factors, there were exactly 400 drivers exhibiting that risk factor and neither of the other two risk factors. Finally, there were exactly 300 drivers who exhibited all three risk factors and 500 who exhibited none of the three risk factors.

Calculate the number of drivers in the study. (A) 2000 (B) 2300 (C) 2450 (D) 2750 (E) 3500

259. An insurance company examines its pool of auto insurance customers and gathers the following information:
i) All customers insure at least one car.
ii) 64% 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 the car is not a sports car.(A) 0.16(B) 0.19(C) 0.26(D) 0.29
(E) 0.31

260. An insurance company has found that 1% of all applicants for life insurance have diabetes.
Calculate the probability that five or fewer of 200 randomly selected applicants have diabetes. (A) 0.85 (B) 0.88 (C) 0.91 (D) 0.95 (E) 0.98

261. The probability that an agent sells an insurance policy to a potential customer during a first appointment is 0.20.

The events of selling an insurance policy to different potential customers during first appointments are mutually independent. The agent has scheduled first appointments with five potential customers.

Calculate the probability that the agent sells an insurance policy during at least two of these appointments. (A) 0.04 (B) 0.20 (C) 0.26 (D) 0.40 (E) 0.74

262. A manufacturer produces computers and releases them in shipments of 100. From a shipment of 100, the probability that exactly three computers are defective is twice the probability that exactly two computers are defective. The events that different computers are defective are mutually independent.

Calculate the probability that a randomly selected computer is defective. (A) 0.040 (B) 0.042 (C) 0.058 (D) 0.060 (E) 0.072

263. In any 12-month period, the probability that a home is damaged by fire is 20% and the probability of a theft loss at a home is 30%. The occurrences of fire damage and theft loss are independent events.

Calculate the probability that a randomly selected home will either be damaged by fire or will have a theft loss, but not both, during the next year. (A) 0.30 (B) 0.38 (C) 0.44 (D) 0.50 (E) 0.56

264. In one company, 30% of males and 20% of females contribute to a supplemental retirement plan. Furthermore, 45% of the company’s employees are female.

Calculate the probability that a randomly selected employee is female, given that this employee contributes to a supplemental retirement plan. (A) 0.09 (B) 0.23 (C) 0.35 (D) 0.45 (E) 0.55

265. A health insurer sells policies to residents of territory X and territory Y. Past claims experience indicates the following:
i) 20% of the total policyholders from territory X and territory Y combined filed no claims.
ii) 15% of the policyholders from territory X filed no claims.
iii) 40% of the policyholders from territory Y filed no claims.
Calculate the probability that a randomly selected policyholder was a resident of territory X, given that the policyholder filed no claims.(A) 0.09(B) 0.27(C) 0.50(D) 0.60(E) 0.80

266. Claim amounts are independent random variables with probability density function

Calculate the probability that the largest of three randomly selected claims is less than 25. (A)(B)(C)(D)(E)

267. The lifetime of a certain electronic device has an exponential distribution with mean 0.50.

Calculate the probability that the lifetime of the device is greater than 0.70, given that it is greater than 0.40. (A) 0.203 (B) 0.247 (C) 0.449 (D) 0.549 (E) 0.861

268. A farmer purchases a five-year insurance policy that covers crop destruction due to hail. Over the five-year period, the farmer will receive a benefit of 20 for each year in which hail destroys his crop, subject to a maximum of three benefit payments. The probability that hail will destroy the farmer’s crop in any given year is 0.5, independent of any other year.

Calculate the expected benefit that the farmer will receive over the five-year period. (A) 30 (B) 34 (C) 40 (D) 46 (E) 50

269. An insurance company has two divisions, auto and property. Total annual claims, X, in the auto division follow a normal distribution with mean 10 and standard deviation 3. Total annual claims, Y, in the property division follow a normal distribution with mean 12 and standard deviation 4.

Assume that X and Y are independent.

Calculate the probability that total overall claims, X + Y, will not exceed 29. (A) 0.61 (B) 0.69 (C) 0.78 (D) 0.84 (E) 0.92

270. An industrial company provides health insurance to employees located at four different plants. Health insurance costs at each plant are independent of the costs at any other plant. Plant managers have calculated the following statistics:

Calculate the standard deviation of total company health insurance costs. (A) 1.4 (B) 2.1 (C) 2.9 (D) 5.5 (E) 6.3

271. In one company, 30% of men and 20% of women contribute to a supplemental retirement plan. Furthermore, 45% of the company’s employees are women.

Calculate the probability that a randomly selected employee is a woman, given that this employee contributes to a supplemental retirement plan. (A) 0.09 (B) 0.23 (C) 0.35 (D) 0.45 (E) 0.55

272. An insurance company has found that 1% of all applicants for life insurance have diabetes.

Calculate the probability that five or fewer of 200 randomly selected applicants have diabetes. (A) 0.85 (B) 0.88 (C) 0.91 (D) 0.95 (E) 0.98

273. An industrial company provides health insurance to employees located at four different plants. Health insurance costs at each plant are independent of the costs at any other plant. Plant managers have calculated the following statistics:

Calculate the standard deviation of total company health insurance costs. (A) 1.4 (B) 2.1 (C) 2.9 (D) 5.5 (E) 6.3

274. An insurance company examines its pool of auto insurance customers and gathers the following information:
i) All customers insure at least one car.
ii) 62% of the customers insure more than one car.
iii) 15% of the customers insure a sports car.
iv) Of those customers who insure more than one car, 20% insure a sports car.

Calculate the probability that a randomly selected customer insures exactly one car, and that the car is not a sports car.(A) 0.230(B) 0.260(C) 0.323(D) 0.354(E) 0.380

275. An insurance company examines its pool of auto insurance customers and gathers the following information:
i) All customers insure at least one car.
ii) 64% 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 the car is not a sports car.(A) 0.16(B) 0.19(C) 0.26(D) 0.29(E) 0.31

276. An insurance company has two divisions, auto and property. Total annual claims, X, in the auto division follow a normal distribution with mean 10 and standard deviation 3. Total annual claims, Y, in the property division follow a normal distribution with mean 12 and standard deviation 4.

Assume that X and Y are independent.

Calculate the probability that total overall claims, X + Y, will not exceed 29. (A) 0.61 (B) 0.69 (C) 0.78 (D) 0.84 (E) 0.92

277. In a study of driver safety, drivers were categorized according to three risk factors. Exactly 1000 drivers exhibited each individual risk factor. Also, for each of the risk factors, there were exactly 400 drivers exhibiting that risk factor and neither of the other two risk factors. Finally, there were exactly 300 drivers who exhibited all three risk factors and 500 who exhibited none of the three risk factors.
.
Calculate the number of drivers in the study. (A) 2000 (B) 2300 (C) 2450 (D) 2750 (E) 3500

278. A farmer purchases a five-year insurance policy that covers crop destruction due to hail. Over the five-year period, the farmer will receive a benefit of 20 for each year in which hail destroys his crop, subject to a maximum of three benefit payments. The probability that hail will destroy the farmer’s crop in any given year is 0.5, independent of any other year.

Calculate the expected benefit that the farmer will receive over the five-year period. (A) 30 (B) 34 (C) 40 (D) 46 (E) 50

279. The lifetime of a certain electronic device has an exponential distribution with mean 0.50.

Calculate the probability that the lifetime of the device is greater than 0.70, given that it is greater than 0.40. (A) 0.203 (B) 0.247 (C) 0.449 (D) 0.549 (E) 0.861

280. The probability that an agent sells an insurance policy to a potential customer during a first appointment is 0.20. The events of selling an insurance policy to different potential customers during first appointments are mutually independent.

The agent has scheduled first appointments with five potential customers.

Calculate the probability that the agent sells an insurance policy during at least two of these appointments. (A) 0.04 (B) 0.20 (C) 0.26 (D) 0.40 (E) 0.74

281. A health insurer sells policies to residents of territory A and territory B. Past claims experience indicates the following:
i) 20% of the total policyholders from territory A and territory B combined filed no claims.
ii) 15% of the policyholders from territory A filed no claims.
iii) 40% of the policyholders from territory B filed no claims.

Calculate the probability that a randomly selected policyholder was a resident of territory A, given that the policyholder filed no claims.(A) 0.09(B) 0.27(C) 0.50(D) 0.60(E) 0.80

283. In any twelve-month period, the probability that a home is damaged by fire is 20% and the probability of a theft loss at a home is 30%. The occurrences of fire damage and theft loss are independent events.

Calculate the probability that a randomly selected home will either be damaged by fire or will have a theft loss, but not both, during the next year. (A) 0.30 (B) 0.38 (C) 0.44 (D) 0.50 (E) 0.56

284. An employer provides disability benefits to its employees for work-related and other injuries. The random variables X and Y denote the employer’s annual expenditures for work-related and other injuries, respectively. An actuarial study reveals the following information about X and Y:
i) The density of X is , for x＞0
ii) Var (Y) = 12,500.
iii) The correlation between X and Y is 0.20.
Calculate the variance of the employer’s total expenditures for work-related and other injuries.(A) 12,500(B) 13,500(C) 15,500(D) 16,500(E) 18,972

285. Appraisals of the value of a necklace are uniformly distributed on the interval [θ−3,θ +1], where θ is the actual price the owner paid for the necklace. Four mutually independent appraisals are obtained.

Let L denote the lowest of the four appraisals and H the highest.

Calculate P[L ＜ θ ＜ H]. (A) 0.152 (B) 0.188 (C) 0.600 (D) 0.680 (E) 0.996

286, Losses follow an exponential distribution with mean 1. Two independent losses are observed.
Calculate the probability that either of the losses is more than twice the other. (A)(B)(C)(D)(E)

287. A manufacturer produces computers and releases them in shipments of 100. From a shipment of 100, the probability that exactly three computers are defective is twice the probability that exactly two computers are defective. The events that different computers are defective are mutually independent.

Calculate the probability that a randomly selected computer is defective. (A) 0.040 (B) 0.042 (C) 0.058 (D) 0.060 (E) 0.072

288. For a pregnant woman, a certain test will give the outcome “not pregnant” with probability 0.10. For a non-pregnant woman, the test will give the outcome “pregnant” with probability 0.20. Of women who take the test, 20% are pregnant.

Calculate the probability that a woman is pregnant, given her test outcome is “pregnant.” (A) 0.10 (B) 0.20 (C) 0.50 (D) 0.53 (E) 0.90

289. An airport owner 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 200 up to a policy maximum of 500.

The following table shows a probability function for the random variable X of winter season snowfall, in inches, at the airport.

Calculate the standard deviation of the amount paid under the policy. (A) 163.5 (B) 187.6 (C) 208.7 (D) 234.9 (E) 336.6

290. Let be independent identically distributed random variables such that P[X=0]= P[X=2] =0.5. Let S= .

Calculate the approximate value of P[S ＞115]. (A) 0.005 (B) 0.067 (C) 0.144 (D) 0.147 (E) 0.440

291. Let X and Y be discrete random variables with joint probability function

Calculate Corr (X, Y), the correlation coefficient of X and Y. (A) 0.06 (B) 0.23 (C) 0.26 (D) 0.38 (E) 0.63

292. Let X and Y be discrete random variables with joint probability function

Calculate the variance of Y. (A) 1.67 (B) 2.22 (C) 3.33 (D) 5.00 (E) 5.56

293. A company provides disability benefits to its employees. There are only two possible benefits: partial disability, costing the company 40, and total disability, costing the company 200. The company employs a number of married couples.
Let (X,Y) denote the company's disability costs for a randomly selected employed married couple. The joint probability function for (X,Y) is:

Calculate the standard deviation of the total disability cost X + Y for the married couple. (A) 12.3 (B) 13.8 (C) 15.7 (D) 16.6 (E) 19.8

294. The probability that the economy will improve, remain stable, or decline is:

Prices for Stock X and Stock Y will change as follows:

Determine which of the following statements about the percentage price changes for Stock X and Stock Y is true. (A) The percentage change for Stock X has a larger variance and a larger mean. (B) The percentage change for Stock X has a larger variance and the means are equal. (C) The percentage change for Stock X has a larger variance and a smaller mean. (D) The variances are equal and the percentage change for Stock X has a larger mean. (E) Both the variances and the means are equal.

295. A company is marketing an investment opportunity to four potential customers. The company believes that its probability of making a sale is 0.5 for each of the first three customers but that it is only 0.1 for the fourth customer. The customers' purchases are independent of one another.

Calculate the probability that at most two customers purchase the investment. (A) 0.38 (B) 0.46 (C) 0.54 (D) 0.84 (E) 0.90

296. An actuary compiles the following information about a portfolio of life insurance policies:
i) There are 150 more policies on males than there are on females.
ii) There are 100 more policies on female nonsmokers than there are on male nonsmokers.
iii) There are 350 policies on smokers.

Calculate the number of policies on female smokers within this portfolio. (A) 50 (B) 100 (C) 200 (D) 250 (E) 300

297. The lifetime of a machine part has a continuous distribution on the interval (0, 40) with probability density function f, where f (x) is proportional to .

Calculate the probability that the lifetime of the machine part is less than five. (A) 0.03 (B) 0.13 (C) 0.42 (D) 0.58 (E) 0.97

298. The claim, X, for a dental insurance policy is a random variable with the following probability function:

The premium for the policy is equal to 125% of the expected claim amount.

Calculate the approximate probability that the total claims on 76 independent policies exceed the total premium collected. (A) 0.02 (B) 0.07 (C) 0.17 (D) 0.25 (E) 0.40

299. An insurance company categorizes its policyholders into three mutually exclusive groups. A study produced the following information:

Within each group, the numbers of claims made by individual policyholders are mutually independent Poisson random variables.

Calculate the expected total number of claims, in thousands, made by the 100,000 policyholders. (A) 21 (B) 28 (C) 36 (D) 64 (E) 72

300. A group of insurance policies have all been in force for at least three years. The insurance company plans to pay a dividend on each policy in the group that had no more than one claim incurred on it in the past three years. The number of claims incurred on a policy in any year follows a Poisson distribution with mean 0.288 and the number incurred in any year is independent of the number incurred in any other year.

Calculate the probability that a policy chosen at random from the group will receive a dividend. (A) 0.01 (B) 0.36 (C) 0.42 (D) 0.54 (E) 0.79

301. An insurance company sells a one-year insurance policy that covers fire and theft losses. The variance of the number of fire losses is 5. The variance of the number of theft losses is 8. The covariance between the number of fire and theft losses is 3.

Calculate the variance of the total number of fire and theft losses covered by this policy. (A) 7 (B) 10 (C) 13 (D) 16 (E) 19

302. The number of automobile accidents on any day in a city has a Poisson distribution with mean 4. The number of accidents on a given day is independent of the number of accidents on any other day.

Calculate the probability that at most one accident occurs in a three-day period. (A)(B) (C) (D)(E)

303. An experiment consists of tossing three fair coins and is deemed a success if the result is three heads or three tails. The experiment is repeated until a success occurs.

Calculate the probability that it takes exactly three experiments to obtain a success. (A) 0.047 (B) 0.070 (C) 0.141 (D) 0.188 (E) 0.422

304. Companies P, Q, and R use routes that take their trucks through a common inspection checkpoint each day. The number of trucks for each company that pass the checkpoint each day is as follows:

Calculate the probability that at least one of two randomly chosen trucks is from Company P. (A) 0.28 (B) 0.31 (C) 0.56 (D) 0.69 (E) 0.72

305. A company administers a typing test to screen applicants for a secretarial position. In order to pass the test, an applicant must complete the test in 50 minutes with no more than one error. Historical data reveals the following about the population of applicants:
i) The number of test errors follows a Poisson distribution with mean 3.
ii) The time required to complete the test follows a normal distribution with mean 45 and standard deviation 10.
iii) The number of errors and the time required to complete the test are independent.
Calculate the probability that an applicant chosen at random will pass the test.(A) 0.10(B) 0.14(C) 0.19(D) 0.84(E) 0.89

306. An insurance company sells 40% of its renters policies to home renters and the remaining 60% to apartment renters. Among home renters, the time from policy purchase until policy cancellation has an exponential distribution with mean 4 years, and among apartment renters, it has an exponential distribution with mean 2 years.

Calculate the probability that the policyholder is a home renter, given that a renter still has a policy one year after purchase. (A) 0.08 (B) 0.27 (C) 0.46 (D) 0.56 (E) 0.66

307. A company sells insurance policies for which benefit payments made to each policyholder are independently and identically normally distributed with mean 2475 and standard deviation 250.

Calculate the minimum number of policies that must be sold for there to be at least a 99% probability that the average benefit paid per policy will be no greater than 2500. (A) 24 (B) 542 (C) 664 (D) 5815 (E) 6440

308. A life insurance policy pays 1000 upon the death of a policyholder provided that the policyholder survives at least one year but less than five years after purchasing the policy.

Let X denote the number of years that a policyholder survives after purchasing the policy with the following probabilities:

Calculate the standard deviation of the payment made under this policy. (A) 218 (B) 430 (C) 480 (D) 495 (E) 500

309. An insurer divides a city into three zones and assesses risks associated with fire loss as follows:

Given that a fire loss occurs in a home covered by the insurer, calculate the probability that the home is in Zone A. (A) 0.349 (B) 0.400 (C) 0.441 (D) 0.465 (E) 0.506

310. An insurer offers policies for which insured loss amounts follow a distribution with density function

Customers may choose one of two policies. Policy 1 has no deductible and a limit of 4, while Policy 2 has a deductible of 4 and no limit.

Given the occurrence of an insured loss, calculate the absolute value of the difference between the insurer’s expected claim payments under Policies 1 and 2. (A) 0.32 (B) 0.64 (C) 0.79 (D) 0.91 (E) 1.12

311. Employees of a large company all choose one of three levels of health insurance coverage, for which premiums, denoted by X , are 1, 2, and 3, respectively. Premiums are subject to a discount, denoted by Y, of 0 for smokers and 1 for non-smokers. The joint probability function of X and Y is given by

Calculate the variance of X Y− , the total premium paid by a randomly chosen employee. (A) 0.20 (B) 0.69 (C) 0.74 (D) 1.90 (E) 2.65

312. An actuary determines the following regarding an individual auto policyholder:
i) The probability that the auto policyholder will file a medical claim is 0.30.
ii) The probability that the auto policyholder will file a property claim is 0.42.
iii) The probability that the auto policyholder will file a medical claim or a property claim is 0.60.

Calculate the probability that the auto policyholder will file exactly one type of claim, given that the policyholder will not file both types of claims. (A) 0.45 (B) 0.48 (C) 0.52 (D) 0.55 (E) 0.60

313. The probability that a homeowners policyholder reports a property claim in a year increases by 25% per year. Conversely, the probability that a homeowners policyholder reports a liability claim in a year decreases by 25% per year.

The probability that a homeowners policyholder reports both a property claim and a liability claim in Year 1 is 0.01. The event that a homeowners policyholder reports a property claim is independent of the event that the policyholder reports a liability claim.

Calculate the probability that a homeowners policyholder reports both a property claim and a liability claim in Year 9. (A) 0.005 (B) 0.006 (C) 0.010 (D) 0.014 (E) 0.015

314. An auto insurance company tracks the experience of its first-year and multi-year policyholders separately. First-year policyholders account for 15% of the company's business while multi-year policyholders account for the rest.

The number of claims reported to the company in a year by a first-year policyholder follows a Poisson distribution with mean 0.50, while the number of claims reported to the company in a year by a multi-year policyholder follows a Poisson distribution with mean 0.20.

Calculate the probability that a policyholder is a first-year policyholder, given that the policyholder reports at least one claim in a year to the company. (A) 0.246 (B) 0.277 (C) 0.306 (D) 0.476 (E) 0.685

315. The random variable characterizes an insurer's annual property losses, where X1 is normally distributed with mean 16 and standard deviation 1.50. Similarly, the random variable characterizes the insurer's annual liability losses, where X2 is normally distributed with mean 15 and standard deviation 2.

The insurer's annual property losses are independent of its annual liability losses.

Calculate the probability that, in a given year, the minimum of the insurer's property losses and liability losses exceeds . (A) 0.126 (B) 0.154 (C) 0.250 (D) 0.309 (E) 0.346

316. A health insurance company classifies applicants, depending on their health, into one of three categories: A, B, or C.

The following probabilities apply:
i) P[A] = 5P[C]
ii) P[B] = 4P[C]
iii) P[ zero claims | A] = 0.1
iv) P[ zero claims | B] = 0.2
v) P[ zero claims | C] = 0.4

Calculate the probability that an insured was classified in category C, given that the insured had zero claims. (A) 0.040 (B) 0.170 (C) 0.235 (D) 0.294 (E) 0.471

317. A five-year term insurance policy pays 25,000 if the insured dies in the first year. The benefit declines by 5000 per year for each of the next four years. In each of the five years covered by the policy, the probability of dying is 0.01, given that the insured is alive at the beginning of that year.

Calculate the expected benefit the insurance company will pay during the five-year term. (A) 692 (B) 740 (C) 750 (D) 985 (E) 1225

318. Data on a certain pregnancy test show that a pregnant woman will test negative or not pregnant 10% of the time, while a non-pregnant woman will test positive 20% of the time.

Thirty percent of the women who take the test are pregnant.

Calculate the probability that a woman is pregnant given that her test outcome is positive. (A) 0.18 (B) 0.30 (C) 0.66 (D) 0.82 (E) 0.90

319. A company is marketing an investment opportunity to four potential customers. The company believes that its probability of making a sale is 0.7 for each of the first three customers but that it is only 0.2 for the fourth customer. The customers' purchases are independent of one another.

Calculate the probability that at most two customers purchase the investment. (A) 0.18 (B) 0.39 (C) 0.57 (D) 0.71 (E) 0.82

112年 - SOCIETY OF ACTUARIES_EXAM P PROBABILITY_EXAM P SAMPLE QUESTIONS 251-319#119927

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