ASA Probability

ASA Online Sample P Exam (Probability)

ASA Online Sample P Exam (Probability)（ASA機率考古題）

  • Question 1 of 30
    • \(A_1\)\(A_2\)\(A_3\)\(A_4\)\(A_5\)
    \(A_6\)\(A_7\)\(A_8\)\(A_9\)\(A_{10}\)
    \(A_{11}\)\(A_{12}\)\(A_{13}\)\(A_{14}\)\(A_{15}\)
    \(A_{16}\)\(A_{17}\)\(A_{18}\)\(A_{19}\)\(A_{20}\)
    \(A_{21}\)\(A_{22}\)\(A_{23}\)\(A_{24}\)\(A_{25}\)
    \(A_{26}\)\(A_{27}\)\(A_{28}\)\(A_{29}\)\(A_{30}\)

    Thirty items are arranged in a 6-by-5 array as shown.

    Calculuate 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.

    1. 200
    2. 760
    3. 1200
    4. 4560
    5. 7200
  • Question 2 of 30

    • 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:

    1. 17% of the clients have none of these three products.
    2. 64% of the clients have auto insurance.
    3. Twice as many of the clients have homeowners insurance as have renters insurance.
    4. 35% of the clients have two of these three products.
    5. 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.

    1. 7%
    2. 10%
    3. 16%
    4. 25%
    5. 28%
  • Question 3 of 30

    • A study is being conducted in which the health of two independent groups of ten policyholders is being monitored over a one-year period of time. Individual participants in the study drop out before the end of the study with probability 0.2 (independently of the other participants).

    What is the probability that at least 9 participants complete the study in one of the two groups, but not in both groups?

    1. 0.096
    2. 0.192
    3. 0.235
    4. 0.376
    5. 0.469
  • Question 4 of 30

    • An auto insurance company has 10,000 policyholders. Each policyholder is classified as

    1. young or old;
    2. male or female; and
    3. married or single.

    Of these policyholders, 3000 are young, 4600 are male, and 7000 are married. The policyholders can also be classified as 1320 young males, 3010 married males, and 1400 young married persons. Finally, 600 of the policyholders are young married males.

    How many of the company's policyholders are young, female, and single?

    1. 280
    2. 423
    3. 486
    4. 880
    5. 896
  • Question 5 of 30

    • 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 are 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.

    1. 0.12
    2. 0.15
    3. 0.80
    4. 0.85
    5. 0.95
  • Question 6 of 30

    • A blood test indicates the presence of a particular disease 95% of the time when the disease is actually present. The same test indicates the presence of the disease 0.5% of the time when the disease is not present. One percent of the population actually has the disease.

    Calculate the probability that a person has the disease given that the test indicates the presence of the disease.

    1. 0.324
    2. 0.657
    3. 0.945
    4. 0.950
    5. 0.995
  • Question 7 of 30

    • A public health researcher examines the medical records of a group of 937 men who died in 1999 and discovers that 210 of the men died from causes related to heart disease. Moreover, 312 of the 937 men had at least one parent who suffered from heart disease, and, of these 312 men,102 died from causes related to heart disease.

    Determine the probability that a man randomly selected from this group died of causes related to heart disease, given that neither of his parents suffered from heart disease.

    1. 0.115
    2. 0.173
    3. 0.224
    4. 0.327
    5. 0.514
  • Question 8 of 30
    • Type of driverPercentage of all driversProbability of at least one collision
    Teen8%0.15
    Young adult16%0.08
    Midlife45%0.04
    Senior31%0.05
    Total100%

    An actuary studied the likelihood that different types of drivers would be involved in at least one collision during any one-year period. The results of the study are presented in the table.

    Given that a driver has been involved in at least one collision in the past year, what is the probability that the driver is a young adult driver?

    1. 0.06
    2. 0.16
    3. 0.19
    4. 0.22
    5. 0.25
  • Question 9 of 30

    • 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.

    1. 0.41
    2. 0.46
    3. 0.58
    4. 0.69
    5. 0.78
  • Question 10 of 30

    • A large pool of adults earning their first driver's license includes 50% 1ow-risk drivers, 30% moderate-risk drivers, and 20% high-risk drivers. Because these drivers have no prior driving record, an insurance company considers each driver to be randomly selected from the pool.

    This month, the insurance company writes 4 new policies for adults earning their first driver's license.

    What is the probability that these 4 will contain at least two more high-risk drivers than low-risk drivers?

    1. 0.006
    2. 0.012
    3. 0.018
    4. 0.049
    5. 0.073
  • Question 11 of 30

    • The distribution of the size of claims paid under an insurance policy has probability density funciton \[ f(x)= \left\{ \begin{array}{ll} cx^a, & 0\lt x\lt 5 \\ 0, & \text{otherwise}, \end{array} \right. \]

    Where \(a\gt 0\) and \(c\gt 0\).

    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.

    1. 0.404
    2. 0.428
    3. 0.500
    4. 0.572
    5. 0.596
  • Question 12 of 30

    • 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.

    1. \(1-F(2)\)
    2. \(F(2)/F(5)\)
    3. \([1-F(2)]/F(5)\)
    4. \([F(0.25)-F(0.1)]/F(0.25)\)
    5. \([F(5)-F(2)]/F(5)\)
  • Question 13 of 30

    • 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:

    1. \(p_n\geq p_{n+1}\) for \(n=0, 1, 2, 3, 4\)
    2. The difference between \(p_n\) and \(p_{n+1}\) is the same for \(n=0, 1, 2, 3, 4\)
    3. 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.

    1. 0.14
    2. 0.16
    3. 0.27
    4. 0.29
    5. 0.33
  • Question 14 of 30

    • Let \(X\), be a continuous random variable with density function \[ f(x)= \left\{ \begin{array}{ll} \frac{|x|}{10} & \text{for }-2\leq x\leq 4 \\ 0 & \text{otherwise} \end{array} \right. \]

    Calculate the expected value of \(X\),

    1. 1/5
    2. 3/5
    3. 1
    4. 28/15
    5. 12/5
  • Question 15 of 30

    • Each time a hurricane arrives, a new home has a 0.4 probability of experiencing damage. The occurrences of damage in different hurricanes are independent.

    Calculate the mode of the number of hurricanes it takes for the home to experience damage from two hurricanes.

    1. 2
    2. 3
    3. 4
    4. 5
    5. 6
  • Question 16 of 30

    • An insurer's annual weather-related loss, \(X\), is a random variable with density function \[ f(x)= \left\{ \begin{array}{ll} \frac{2.5(200)^{2.5}}{x^{3.5}} & \text{for }x\gt 200 \\ 0 & \text{otherwise} \end{array} \right. \]

    Calculate the difference between the 30th and 70th percentiles of \(X\).

    1. 35
    2. 93
    3. 124
    4. 231
    5. 298
  • Question 17 of 30

    • 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.

    1. 554
    2. 872
    3. 1022
    4. 1354
    5. 1612
  • Question 18 of 30

    • 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 \[ f(x)= \left\{ \begin{array}{ll} 1/5 & \text{for }0\lt x\lt 5 \\ 0 & \text{otherwise} \end{array} \right. \] Let \(Y\) be the age of the machine at the time of replacement.

    Determine the variance of \(Y\).

    1. 1.3
    2. 1.4
    3. 1.7
    4. 2.1
    5. 7.5
  • Question 19 of 30

    • An insurance policy on an electrical device pays a benefit of 4000 if the device fails during the first year. The amount of the benefit decreases by 1000 each successive year until it reaches 0. If the device has not failed by the beginning of any given year, the probability of failure during that year is 0.4.

    What is the expected benefit under this policy?

    1. 2234
    2. 2400
    3. 2500
    4. 2667
    5. 2694
  • Question 20 of 30

    • 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.

    1. 0.48
    2. 0.53
    3. 0.54
    4. 0.67
    5. 0.68
  • Question 21 of 30

    • The proportion \(X\) of yearly dental claims that exceed 200 is a random variable with probability density function \[ f(x)= \left\{ \begin{array}{ll} 60x^3(1-x)^2, & 0\lt x\lt 1 \\ 0, & \text{otherwise}. \end{array} \right. \]

    Calculate \(\text{Var}[X/(1-X)]\)

    1. 149/900
    2. 10/7
    3. 6
    4. 8
    5. 10
  • Question 22 of 30

    • 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.

    1. 0.2857
    2. 0.3214
    3. 0.3333
    4. 0.3571
    5. 0.4000
  • Question 23 of 30

    • 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.

    What is 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?

    1. \(\frac{1}{18}(1-e^{-2/3}-e^{-1/2}+e^{-7/6})\)
    2. \(\frac{1}{18}e^{-7/6}\)
    3. \(1-e^{-2/3}-e^{-1/2}+e^{-7/6}\)
    4. \(1-e^{-2/3}-e^{-1/2}+e^{-1/3}\)
    5. \(1-\frac{1}{3}e^{-2/3}-\frac{1}{6}e^{-1/2}+\frac{1}{18}e^{-7/6}\)
  • Question 24 of 30

    • Two life insurance policies, each with a death benefit of 10,000 and a one-time premium of 500, are sold to a 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 is 0.025, 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.

    What is the expected excess of premiums over claims, given that the husband survives at least ten years?

    1. 350
    2. 385
    3. 397
    4. 870
    5. 897
  • Question 25 of 30

    • 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.

    1. 1/5
    2. 7/15
    3. 3/5
    4. 11/15
    5. 1
  • Question 26 of 30

    • 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 \[ p(x,y)= \left\{ \begin{array}{ll} c(x+2y), & \text{for }x=0,1,2,3, y=0,1,2,3, x\leq y \\ 0, & \text{otherwise}, \end{array} \right. \]

    where \(c\) is a constant.

    Calculate the expected number of tornadoes that result in fewer than 50 million in losses.

    1. 0.19
    2. 0.28
    3. 0.76
    4. 1.00
    5. 1.10
  • Question 27 of 30

    • 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:

    Claim Amount
    1005001000
    Territory 10.900.080.02
    Territory 20.800.110.09
    Territory 30.700.200.10

    Calculate the standard deviation of a randomly selected claim amount.

    1. 254
    2. 291
    3. 332
    4. 368
    5. 396
  • Question 28 of 30

    • 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.

    1. 4,000,000
    2. 4,004,000
    3. 8,000,000
    4. 16,000,000
    5. 20,000,000
  • Question 29 of 30

    • 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 independent lifetimes.

    What is 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?

    1. 14
    2. 16
    3. 20
    4. 40
    5. 55
  • Question 30 of 30

    • Two instruments are used to measure the height, \(h\), of a tower. The error made by the less accurate instrument is normally distributed with mean 0 and standard deviation \(0.0056h\). The error made by the more accurate instrument is normally distributed with mean 0 and standard deviation \(0.0044h\).

    Assuming the two measurements are independent random variables, what is the probability that their average value is within \(0.005h\) of the height of the tower?

    1. 0.38
    2. 0.40
    3. 0.68
    4. 0.84
    5. 0.90

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