112年 - SOCIETY OF ACTUARIES_EXAM P PROBABILITY_EXAM P SAMPLE QUESTIONS 201-250#119925

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. (A)(B)(C)(D)(E)

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

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. (A)

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

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

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

Calculate the probability that the loss on a randomly selected claim is greater than the value of the automobile. (A) 0.00 (B) 0.10 (C) 0.25 (D) 0.75 (E) 0.90

208. The lifetime of a machine part is exponentially distributed with a mean of five years.

Calculate the mean lifetime of the part, given that it survives less than ten years. (A) 0.865 (B) 1.157 (C) 2.568 (D) 2.970 (E) 3.435

209. Let X be a random variable with density function

Calculate P [X≤0.5|X≤1.0]. (A) 0.433 (B) 0.547 (C) 0.632 (D) 0.731 (E) 0.865

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, is a random variable with P[N=n]= , for n=0,1,2,.... The numbers of claims 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. (A)(B)r³ (C)(D)r² (E)

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

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

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. (A) 392 (B) 512 (C) 672 (D) 882 (E) 1008

221. An insurance policy will reimburse only one claim per year.

For a random policyholder, there is a 20% probability of no loss in the next year, in which case the claim amount is 0. If a loss occurs in the next year, the claim amount is normally distributed with mean 1000 and standard deviation 400.

Calculate the median claim amount in the next year for a random policyholder. (A) 663 (B) 790 (C) 873 (D) 994 (E) 1000

222. Losses incurred by a policyholder follow a normal distribution with mean 20,000 and standard deviation 4,500. The policy covers losses, subject to a deductible of 15,000.

Calculate the percentile of losses that exceed the deductible. (A) 27,400 (B) 27,700 (C) 28,100 (D) 28,400 (E) 28,800

223. A gun shop sells gunpowder. Monthly demand for gunpowder is normally distributed, averages 20 pounds, and has a standard deviation of 2 pounds. The shop manager wishes to stock gunpowder inventory at the beginning of each month so that there is only a 2% chance that the shop will run out of gunpowder (i.e., that demand will exceed inventory) in any given month.

Calculate the amount of gunpowder to stock in inventory, in pounds. (A) 16 (B) 23 (C) 24 (D) 32 (E) 43

224. A large university will begin a 13-day period during which students may register for that semester’s courses. Of those 13 days, the number of elapsed days before a randomly selected student registers has a continuous distribution with density function f(t) that is symmetric about t = 6.5 and proportional to 1/(t + 1) between days 0 and 6.5.

A student registers at the 60th percentile of this distribution.

Calculate the number of elapsed days in the registration period for this student. (A) 4.01 (B) 7.80 (C) 8.99 (D) 10.22 (E) 10.51

225. The loss L due to a boat accident is exponentially distributed.

Boat insurance policy A covers up to 1 unit for each loss. Boat insurance policy B covers up to 2 units for each loss.

The probability that a loss is fully covered under policy B is 1.9 times the probability that it is fully covered under policy A.

Calculate the variance of L. (A) 0.1 (B) 0.4 (C) 2.4 (D) 9.5 (E) 90.1

226. Losses, X, under an insurance policy are exponentially distributed with mean 10. For each loss, the claim payment Y is equal to the amount of the loss in excess of a deductible d ＞ 0.
Calculate Var(Y). (A)100 - d (B)(10− d)² (C)(D) (E)

227. For a certain insurance company, 10% of its policies are Type A, 50% are Type B, and 40% are Type C.
The annual number of claims for an individual Type A, Type B, and Type C policy follow Poisson distributions with respective means 1, 2, and 10.

Let X represent the annual number of claims of a randomly selected policy.

Calculate the variance of X. (A) 5.10 (B) 16.09 (C) 21.19 (D) 42.10 (E) 47.20

228. The number of tornadoes in a given year follows a Poisson distribution with mean 3.
Calculate the variance of the number of tornadoes in a year given that at least one tornado occurs. (A) 1.63 (B) 1.73 (C) 2.66 (D) 3.00 (E) 3.16

229. A government employee’s yearly dental expense follows a uniform distribution on the interval from 200 to 1200. The government’s primary dental plan reimburses an employee for up to 400 of dental expense incurred in a year, while a supplemental plan pays up to 500 of any remaining dental expense.

Let Y represent the yearly benefit paid by the supplemental plan to a government employee.

Calculate Var(Y). (A) 20,833 (B) 26,042 (C) 41,042 (D) 53,333 (E) 83,333

230. Under a liability insurance policy, losses are uniformly distributed on [0, b], where b is a positive constant. There is a deductible of b/2.

Calculate the ratio of the variance of the claim payment (greater than or equal to zero) from a given loss to the variance of the loss. (A) 1:8 (B) 3:16 (C) 1:4 (D) 5:16 (E) 1:2

231. A company’s annual profit, in billions, has a normal distribution with variance equal to the cube of its mean. The probability of an annual loss is 5%.

Calculate the company’s expected annual profit. (A) 370 million (B) 520 million (C) 780 million (D) 950 million (E) 1645 million

232. The number of claims X on a health insurance policy is a random variable with
E[X²]=61 and E[X-1]²=47.
Calculate the standard deviation of the number of claims. (A) 2.18 (B) 2.40 (C) 7.31 (D) 7.50 (E) 7.81

233. Ten cards from a deck of playing cards are in a box: two diamonds, three spades, and five hearts. Two cards are randomly selected without replacement.

Calculate the variance of the number of diamonds selected, given that no spade is selected. (A) 0.24 (B) 0.28 (C) 0.32 (D) 0.34 (E) 0.41

237. A car and a bus arrive at a railroad crossing at times independently and uniformly distributed between 7:15 and 7:30. A train arrives at the crossing at 7:20 and halts traffic at the crossing for five minutes.

Calculate the probability that the waiting time of the car or the bus at the crossing exceeds three minutes. (A) 0.25 (B) 0.27 (C) 0.36 (D) 0.40 (E) 0.56

238. Skateboarders A and B practice one difficult stunt until becoming injured while attempting the stunt. On each attempt, the probability of becoming injured is p, independent of the outcomes of all previous attempts.

Let F(x, y) represent the probability that skateboarders A and B make no more than x and y attempts, respectively, where x and y are positive integers.

It is given that F(2, 2) = 0.0441.

Calculate F(1, 5). (A) 0.0093 (B) 0.0216 (C) 0.0495 (D) 0.0551 (E) 0.1112

239. The number of minor surgeries, X, and the number of major surgeries, Y, for a policyholder, this decade, has joint cumulative distribution function

for nonnegative integers x and y.

Calculate the probability that the policyholder experiences exactly three minor surgeries and exactly three major surgeries this decade. (A) 0.00004 (B) 0.00040 (C) 0.03244 (D) 0.06800 (E) 0.12440

240. A company provides a death benefit of 50,000 for each of its 1000 employees. There is a 1.4% chance that any one employee will die next year, independent of all other employees. The company establishes a fund such that the probability is at least 0.99 that the fund will cover next year’s death benefits.

Calculate, using the Central Limit Theorem, the smallest amount of money, rounded to the nearest 50 thousand, that the company must put into the fund. (A) 750,000 (B) 850,000 (C) 1,050,000 (D) 1,150,000 (E) 1,400,000

241. An investor invests 100 dollars in a stock. Each month, the investment has probability 0.5 of increasing by 1.10 dollars and probability 0.5 of decreasing by 0.90 dollars. The changes in price in different months are mutually independent.

Calculate the probability that the investment has a value greater than 91 dollars at the end of month 100. (A) 0.63 (B) 0.75 (C) 0.82 (D) 0.94 (E) 0.97

242. Let X denote the loss amount sustained by an insurance company’s policyholder in an auto collision. Let Z denote the portion of X that the insurance company will have to pay. An actuary determines that X and Z are independent with respective density and probability functions

Calculate the variance of the insurance company’s claim payment ZX. (A) 13.0 (B) 15.8 (C) 28.8 (D) 35.2 (E) 44.6

243. A couple takes out a medical insurance policy that reimburses them for days of work missed due to illness. Let X and Y denote the number of days missed during a given month by the wife and husband, respectively. The policy pays a monthly benefit of 50 times the maximum of X and Y, subject to a benefit limit of 100. X and Y are independent, each with a discrete uniform distribution on the set {0,1,2,3,4}.

Calculate the expected monthly benefit for missed days of work that is paid to the couple. (A) 70 (B) 90 (C) 92 (D) 95 (E) 140

244. The table below shows the joint probability function of a sailor’s number of boating accidents and number of hospitalizations from these accidents this year.
Calculate the sailor’s expected number of hospitalizations from boating accidents this year. (A) 0.085 (B) 0.099 (C) 0.410 (D) 1.000 (E) 1.500

245. On Main Street, a driver’s speed just before an accident is uniformly distributed on [5, 20]. Given the speed, the resulting loss from the accident is exponentially distributed with mean equal to three times the speed.

Calculate the variance of a loss due to an accident on Main Street. (A) 525 (B) 1463 (C) 1575 (D) 1632 (E) 1744

246. Let X be the annual number of hurricanes hitting Florida, and let Y be the annual number of hurricanes hitting Texas. X and Y are independent Poisson variables with respective means 1.70 and 2.30.

Calculate Var(X−Y| X+Y=3). (A) 1.71 (B) 1.77 (C) 2.93 (D) 3.14 (E) 4.00

247. Random variables X and Y have joint distribution

Let a be the value that minimizes the variance of X.
Calculate the variance of Y. (A) 2/5 (B) 8/15 (C) 16/25 (D) 2/3 (E) 7/10

248. Let X be a random variable that takes on the values –1, 0, and 1 with equal probabilities.
Let Y=X² .
Which of the following is true? (A) Cov(X, Y) ＞ 0； the random variables X and Y are dependent. (B) Cov(X, Y) ＞ 0； the random variables X and Y are independent. (C) Cov(X, Y) = 0； the random variables X and Y are dependent. (D) Cov(X, Y) = 0； the random variables X and Y are independent. (E) Cov(X, Y) ＜ 0； the random variables X and Y are dependent.

249. Losses follow an exponential distribution with mean 1. Two independent losses are observed.

Calculate the expected value of the smaller loss. (A) 0.25 (B) 0.50 (C) 0.75 (D) 1.00 (E) 1.50

250. A delivery service owns two cars that consume 15 and 30 miles per gallon. Fuel costs 3 per gallon. On any given business day, each car travels a number of miles that is independent of the other and is normally distributed with mean 25 miles and standard deviation 3 miles.

Calculate the probability that on any given business day, the total fuel cost to the delivery service will be less than 7. (A) 0.13 (B) 0.23 (C) 0.29 (D) 0.38 (E) 0.47

112年 - SOCIETY OF ACTUARIES_EXAM P PROBABILITY_EXAM P SAMPLE QUESTIONS 201-250#119925

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