112年 - SOCIETY OF ACTUARIES_EXAM P PROBABILITY_EXAM P SAMPLE QUESTIONS 101-150#119921

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)

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%
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 percentile of the insurance company payout on this policy. (A) 3466 (B) 3659 (C) 4159 (D) 8487 (E) 8987
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

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 
 , where Ri is the area of the red sector.
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

110. The probability of x losses occurring in year 1 is 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:

 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

 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.

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

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)(B)(C)(D)(E)
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

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. 
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

 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
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
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
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
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

 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

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. (A)(B)(C)(D)(E)
132. A company issues auto insurance policies. There are 900 insured individuals. Fifty-four percent of them are male. If a female is randomly selected from the 900, the probability she is over 25 years old is 0.43. There are 395 total insured individuals over 25 years old.
A person under 25 years old is randomly selected.
Calculate the probability that the person selected is male. (A) 0.47 (B) 0.53 (C) 0.54 (D) 0.55 (E) 0.56

133. An insurance company insures red and green cars. An actuary compiles the following data:
  

The actuary randomly picks a claim from all claims that exceed the deductible.
Calculate the probability that the claim is on a red car. (A) 0.300 (B) 0.462 (C) 0.491 (D) 0.667 (E) 0.692
134. George and Paul play a betting game. Each chooses an integer from 1 to 20 (inclusive) at random. If the two numbers differ by more than 3, George wins the bet. Otherwise, Paul wins the bet.
Calculate the probability that Paul wins the bet. (A) 0.27 (B) 0.32 (C) 0.40 (D) 0.48 (E) 0.66
135. A student takes an examination consisting of 20 true-false questions. The student knows the answer to N of the questions, which are answered correctly, and guesses the answers to the rest. The conditional probability that the student knows the answer to a question, given that the student answered it correctly, is 0.824..
Calculate N. (A) 8 (B) 10 (C) 14 (D) 16 (E) 18
136. The minimum force required to break a particular type of cable is normally distributed with mean 12,432 and standard deviation 25. A random sample of 400 cables of this type is selected.
Calculate the probability that at least 349 of the selected cables will not break under a force of 12,400. (A) 0.62 (B) 0.67 (C) 0.92 (D) 0.97 (E) 1.00
137. The number of policies that an agent sells has a Poisson distribution with modes at 2 and3.
K is the smallest number such that the probability of selling more than K policies is less than 25%.
Calculate K. (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
138. Two fair dice are tossed. One die is red and one die is green.

Calculate the probability that the sum of the numbers on the two dice is an odd number given that the number that shows on the red die is larger than the number that shows on the green die. (A) 1/4 (B) 5/12 (C) 3/7 (D) 1/2 (E) 3/5
139. In 1982 Abby’s mother scored at the 93rd percentile in the math SAT exam. In 1982 the mean score was 503 and the variance of the scores was 9604.

In 2008 Abby took the math SAT and got the same numerical score as her mother had received 26 years before. In 2008 the mean score was 521 and the variance of the scores was 10,201.

Math SAT scores are normally distributed and stated in multiples of ten.

Calculate the percentile for Abby’s score. (A) 89th (B) 90th (C) 91st (D) 92nd (E) 93rd
140. A certain brand of refrigerator has a useful life that is normally distributed with mean 10 years and standard deviation 3 years. The useful lives of these refrigerators are independent.

Calculate the probability that the total useful life of two randomly selected refrigerators will exceed 1.9 times the useful life of a third randomly selected refrigerator. (A) 0.407 (B) 0.444 (C) 0.556 (D) 0.593 (E) 0.604
141. Losses covered by a flood insurance policy are uniformly distributed on the interval [0, 2]. The insurer pays the amount of the loss in excess of a deductible d.

The probability that the insurer pays at least 1.20 on a random loss is 0.30.

Calculate the probability that the insurer pays at least 1.44 on a random loss. (A) 0.06 (B) 0.16 (C) 0.18 (D) 0.20 (E) 0.28

142. The lifespan, in years, of a certain computer is exponentially distributed. The probability that its lifespan exceeds four years is 0.30. Let f(x) represent the density function of the computer’s lifespan, in years, for x ＞ 0. Determine which of the following is an expression for f(x). (A)(B)(C)(D)(E)

143. The lifetime of a light bulb has density function, f, where f(x) is proportional to , 0<x<5 ,and 0, otherwise.
Calculate the mode of this distribution. (A) 0.00 (B) 0.79 (C) 1.26 (D) 4.42 (E) 5.00

144. An insurer’s medical reimbursements have density function f, where f(x) is proportional to , for 0 ＜ x ＜ 1, and 0, otherwise.
Calculate the mode of the medical reimbursements. (A) 0.00 (B) 0.50 (C) 0.71 (D) 0.84 (E) 1.00
145. A company has five employees on its health insurance plan. Each year, each employee independently has an 80% probability of no hospital admissions. If an employee requires one or more hospital admissions, the number of admissions is modeled by a geometric distribution with a mean of 1.50. The numbers of hospital admissions of different employees are mutually independent.

Each hospital admission costs 20,000.

Calculate the probability that the company’s total hospital costs in a year are less than 50,000. (A) 0.41 (B) 0.46 (C) 0.58 (D) 0.69 (E) 0.78
146. On any given day, a certain machine has either no malfunctions or exactly one malfunction. The probability of malfunction on any given day is 0.40. Machine malfunctions on different days are mutually independent.

Calculate the probability that the machine has its third malfunction on the fifth day, given that the machine has not had three malfunctions in the first three days. (A) 0.064 (B) 0.138 (C) 0.148 (D) 0.230 (E) 0.246
147. In a certain group of cancer patients, each patient's cancer is classified in exactly one of the following five stages: stage 0, stage 1, stage 2, stage 3, or stage 4.
i) 75% of the patients in the group have stage 2 or lower.
ii) 80% of the patients in the group have stage 1 or higher.
iii) 80% of the patients in the group have stage 0, 1, 3, or 4.
One patient from the group is randomly selected.

Calculate the probability that the selected patient's cancer is stage 1.(A) 0.20(B) 0.25(C) 0.35(D) 0.48(E) 0.65
148. A car is new at the beginning of a calendar year. The time, in years, before the car experiences its first failure is exponentially distributed with mean 2.

Calculate the probability that the car experiences its first failure in the last quarter of some calendar year. (A) 0.081 (B) 0.088 (C) 0.102 (D) 0.205 (E) 0.250
149. In a shipment of 20 packages, 7 packages are damaged. The packages are randomly inspected, one at a time, without replacement, until the fourth damaged package is discovered.

Calculate the probability that exactly 12 packages are inspected. (A) 0.079 (B) 0.119 (C) 0.237 (D) 0.243 (E) 0.358

150. A theme park conducts a study of families that visit the park during a year. The fraction of such families of size m is , m = 1, 2, 3, 4, 5, 6, and 7.

For a family of size m that visits the park, the number of members of the family that ride the roller coaster follows a discrete uniform distribution on the set {1, … , m}.

Calculate the probability that a family visiting the park has exactly six members, given that exactly five members of the family ride the roller coaster. (A) 0.17 (B) 0.21 (C) 0.24 (D) 0.28 (E) 0.31

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