Tag Archives: Binomial distribution

Exam P Practice Problem 76 – Quantifying Average Random Loss

Both Problem 76-A and Problem 76-B use the following information.

A property owner faces a series of independent random losses. Each loss is either 10 (with probability 0.4) or 50 (with probability 0.6).

Three such random losses are selected.

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Problem 76-A

What is the probability that the mean of the three losses is less than 30?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.06

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.32

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.35

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.43

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Problem 76-B

What is the expected value of the mean of the three losses?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 30

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 32

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 33

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 34

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 35

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Exam P Practice Problem 71 – Estimating Claim Frequency

Problem 71-A

An auto insurer issued policies to a large group of drivers under the age of 40. These drivers are classified into five distinct groups by age. These groups are equal in size.

The annual claim count distribution for any driver being insured by this insurer is assumed to be a binomial distribution. The following table shows more information about these drivers.

      \displaystyle \begin{bmatrix} \text{Age}&\text{ }&\text{ }&\text{Mean} &\text{ }&\text{ }&\text{Variance} \\\text{Group}&\text{ }&\text{ }&\text{Of Claim Count} &\text{ }&\text{ }&\text{Of Claim Count} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{16-17}&\text{ }&\text{ }&\displaystyle \frac{5}{2}&\text{ }&\text{ }&\displaystyle \frac{5}{4} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{18-24}&\text{ }&\text{ }&\displaystyle 2&\text{ }&\text{ }&\displaystyle 1 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{25-29}&\text{ }&\text{ }&\displaystyle \frac{3}{2}&\text{ }&\text{ }&\displaystyle \frac{3}{4} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{30-34}&\text{ }&\text{ }&\displaystyle 1&\text{ }&\text{ }&\displaystyle \frac{1}{2} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{35-39}&\text{ }&\text{ }&\displaystyle \frac{1}{2} &\text{ }&\text{ }&\displaystyle \frac{1}{4}    \end{bmatrix}

An insured driver is randomly selected from this large pool of insured and is observed to have one claim in the last year.

What is the probability that the mean number of claims in a year for this insured driver is greater than 1.5?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{14}{67}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{13}{57}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{3}{5}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{51}{67}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{50}{64}

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Problem 71-B

An auto insurer issued policies to a large group of drivers under the age of 40. These drivers are classified into five distinct groups by age. These groups are equal in size.

The annual claim count distribution for any driver being insured by this insurer is assumed to be a geometric distribution. The following table shows more information about these drivers.

      \displaystyle \begin{bmatrix} \text{Age}&\text{ }&\text{ }&\text{Mean} &\text{ }&\text{ }&\text{Variance} \\\text{Group}&\text{ }&\text{ }&\text{Of Claim Count} &\text{ }&\text{ }&\text{Of Claim Count} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{35-39}&\text{ }&\text{ }&\displaystyle 1 &\text{ }&\text{ }&\displaystyle 2 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{30-34}&\text{ }&\text{ }&\displaystyle 2&\text{ }&\text{ }&\displaystyle 6 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{25-29}&\text{ }&\text{ }&\displaystyle 3&\text{ }&\text{ }&\displaystyle 12 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{18-24}&\text{ }&\text{ }&\displaystyle 4&\text{ }&\text{ }&\displaystyle 20 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{16-17}&\text{ }&\text{ }&\displaystyle 5&\text{ }&\text{ }&\displaystyle 30    \end{bmatrix}

An insured driver is randomly selected from this large pool of insured and is observed to have one claim in the last year.

What is the probability that the mean number of claims in a year for this insured driver is greater than 2.5?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.49

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.51

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.55

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.57

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.60

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Exam P Practice Problem 69 – More Large Claim Studies

Problem 69-A

The size of a claim (in thousands) arising from a large portfolio of property and casualty insurance policies is modeled by the random variable X. The following is the probability density function of X.

      \displaystyle f(x)=\frac{0.05^8}{5040} \ x^7 \ e^{-0.05 \ x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x>0

The size of a claim is independent of the size of any other claim in this portfolio of insurance policies.

An actuary is hired to study the large claims arising from these insurance policies, in particular, any claim size greater than the 80th percentile of the claim size distribution.

In a random sample of 10 claims from this portfolio of insurance policies, what is the probability that more than two of the claims are considered large by this actuary?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3020

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3222

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6242

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6778

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6980

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Problem 69-B

The size of a claim (in thousands) arising from a large portfolio of property and casualty insurance policies is modeled by the random variable X. The following is the probability density function of X.

      \displaystyle f(x)=\frac{0.04^{11}}{10!} \ x^{10} \ e^{-0.04 x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x>0

The size of a claim is independent of the size of any other claim in this portfolio of insurance policies.

An actuary is hired to study the large claims arising from these insurance policies, in particular, any claim size greater than the 90th percentile of the claim size distribution.

In a random sample of 15 claims from this portfolio of insurance policies, what is the probability that two or more of the claims are considered large by this actuary?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.184

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.267

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.451

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.733

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.816

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Exam P Practice Problem 65 – Total Insurance Payment

Problem 65-A

The number of random losses in a calendar year for an individual has a Poisson distribution with mean 1. When a loss occurs, the individual loss amount is either 2 or 4, with probabilities 0.6 and 0.4, respectively.

When multiple losses occur for this individual, the individual loss amounts are independent.

This individual purchases an insurance policy to cover the random losses with a deductible of 1 per loss.

In the next calendar year, let S be the total payment made by the insurance company to the insured. Calculate P(2 \le S \le 4).

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12204

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.16974

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29380

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.31689

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34277

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Problem 65-B

The number of claims in a calendar year for an insured has a Poisson distribution with mean 1.2. When a claim occurs, the individual claim amount is either 10 or 20, with probabilities 0.8 and 0.2, respectively.

When multiple claims occur for this insured, the individual claim amounts are independent.

This individual purchases an insurance policy to cover the random losses with a deductible of 5 per loss.

In the next calendar year, let S be the total payment made by the insurance company to the insured. Calculate P(10 \le S < 30).

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1010

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2986

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3709

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3826

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3906

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Exam P Practice Problem 56 – Reporting of Auto Accidents

Problem 56-A

An insurer sells auto insurance policies that provide collision coverage to drivers. The collision accidents reported by drivers are uniformly distributed across the days of the week.

The day of reporting an accident is independent of the day of reporting of any other accident.

Suppose that in one week, 10 collision accidents are reported to the insurer. What is the probability that more than 3 accidents are reported on Saturday and Sunday?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1269

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3127

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.4218

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.5782

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6873

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Problem 56-B

An insurer sells auto insurance policies that provide collision coverage to drivers. The collision accidents reported by drivers are uniformly distributed across the days of the week.

The day of reporting an accident is independent of the day of reporting of any other accident. The number of accidents reported in one week is also independent of the number of accidents reported in any other week.

Suppose that in one week, 10 collision accidents are reported to the insurer and in the following week, 12 collision accidents are reported to the insurer. What is the probability that more than 20% of the accidents from these two weeks are reported on Saturday and Sunday?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0571

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0886

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2028

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.7972

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.9114

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Exam P Practice Problem 47 – A Study of Cardiovascular Patients

Problem 47-A

Ten participants are randomly selected from a population of patients with history of heart disease to be tracked in a health study. The study begins with an initial assessment of health status. The participants are instructed to return for a follow up visit one year after the initial assessment.

For these ten participants, the time (in years) from the initial assessment to the first heart attack has an exponential distribution with mean 5.

The times to first heart attack for these participants are independent.

What is the probability that prior to the one-year follow up visit, three or more of the participants have experienced their first heart attack since initial assessment?

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Problem 47-B

The time (in years) to failure of an electronic device follows an exponential distribution with mean 10. Twelve such devices are randomly selected for a quality control study.

The times to failure for these devices are independent.

What is the probability that at least 9 of the devices will still be working after 5 years?

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Exam P Practice Problem 44 – Traffic Statistics Using Poisson Distribution

Problem 44-A

The number of car accidents in a stretch of a highway (Highway #1) has a Poisson distribution with a mean of 4 per week. The number of car accidents in a stretch of another highway (Highway #2) has a Poisson distribution with a mean of 8 per week.

Assume that on a weekly basis, the number of accidents in one highway is independent of the number of accidents in the other highway.

In one particular week, exactly 5 auto accidents took place in these two highways. What is the probability that Highway #1 had exactly 2 accidents in this particular week?

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Problem 44-B

The number of cars (in a day) that break down in a stretch of a highway (Highway #1) has a Poisson distribution with a mean of 16. The number of cars (in a day) that break down in a stretch of another highway (Highway #2) has a Poisson distribution with a mean of 8.

Assume that on a daily basis, the number of cars breaking down in one highway is independent of the number of cars breaking down in the other highway.

In one particular day, exactly 8 cars were found to break down in these two highways. What is the probability that Highway #1 had exactly 5 cars breaking down in this particular day?

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Exam P Practice Problem 42 – Losses due to Collisions

Problem 42-A

Amount of damage to cars (in thousands) due to collision accidents is distributed according to the following density function:

      \displaystyle f(x)=\frac{1}{8} \ (4-x) \ \ \ \ \ \ 0<x<4

Five independent collision accidents are randomly selected. What is the probability that the maximum amount of damage from these five accidents is greater than 2?

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Problem 42-B

A certain type of random losses is distributed according to the following density function:

      \displaystyle f(x)=\frac{1}{18} \ x \ \ \ \ \ \ 0<x<6

Three independent loss amounts are randomly selected. What is the probility that the minimum amount of these losses is no greater than 3?

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Exam P Practice Problem 41 – Conditional Expected Number of Balls

Problem 41-A

An insurer has a block of business where the number of claims in a year for a policyholder in the block has the following probability distribution.

            \displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Probability}  \\\text{ }&\text{ }&\text{ } \\ 0&\text{ }&\displaystyle 0.08 \\\text{ }&\text{ }&\text{ }  \\ 1&\text{ }&\displaystyle 0.35 \\\text{ }&\text{ }&\text{ }  \\ 2&\text{ }&\displaystyle 0.40 \\\text{ }&\text{ }&\text{ }  \\ 3&\text{ }&\displaystyle 0.12 \\\text{ }&\text{ }&\text{ }  \\ 4&\text{ }&\displaystyle 0.0375 \\\text{ }&\text{ }&\text{ }  \\ 5&\text{ }&\displaystyle 0.0125   \end{bmatrix}

Two policyholders from this block are randomly selected and observed for a year. It is found that there are exactly four claims for these two policyholders in the past year. What is the probability that one of the policyholders has exactly three claims?

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Problem 41-B

There are two bowls containing red balls and white balls. Bowl 1 contains 5 red balls and 5 white balls. Five balls are randomly selected from Bowl 1 without replacement. Bowl 2 also contains 5 red balls and 5 white balls. Five balls are randomly selected from Bowl 2 with replacement.

If the total number of red balls drawn from the two bowls is eight, what is the expected number of red balls from Bowl 2?

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Exam P Practice Problem 40 – Total Claim Amount

Problem 40-A

The number of claims in a calendar year for an insured has a probability function indicated below.

            \displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Probability}  \\\text{ }&\text{ }&\text{ } \\ 0&\text{ }&\displaystyle \frac{27}{64} \\\text{ }&\text{ }&\text{ }  \\ 1&\text{ }&\displaystyle \frac{27}{64} \\\text{ }&\text{ }&\text{ }  \\ 2&\text{ }&\displaystyle \frac{9}{64} \\\text{ }&\text{ }&\text{ }  \\ 3&\text{ }&\displaystyle \frac{1}{64}     \end{bmatrix}

When a claim occurs, the claim amount X, regardless of how many claims the insured will have in the calendar year, has probabilities P(X=1)=0.8 and P(X=2)=0.2. The claim amounts in a calendar year for this insured are independent.

Let T be the total claim amount for this insured in a calendar year. Calculate P(3 \le T \le 4).

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Problem 40-B

A bowl has 3 red balls and 6 white balls. Select two balls at random from this bowl with replacement. Let N be the number of red balls found in the two selected balls. When N=n where n>0, roll a fair die n times.

Let W be the sum of the rolls of the die. Calculate P(4 \le W \le 5).

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