Category Archives: Insurance and Risk Management

Exam P Practice Problem 88 – Expected Value of Insurance Payments

By Dan Ma on July 3, 2014 | Leave a comment

Problem 88-A

A random loss $X$ has a uniform distribution over the interval $0<x<20$ . An insurance policy is purchased to reimburse the loss up to a maximum limit of $m$ where $0<m<20$ .

The expected value of the benefit payment under this policy is 8.4. Calculate the value of $m$ .

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 8.7$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 9.0$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 12.0$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 13.6$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 18.3$

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Problem 88-B

An individual purchases an insurance policy to cover a loss $X$ whose density function is:

$\displaystyle f(x)=\frac{2}{25} \ (5-x) \ \ \ \ \ \ \ \ 0<x<5$

The insurance policy reimburses the policy owner up to a benefit limit of 4 for each loss. What is the expected value of insurance payment made to the policy owner?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 1.35$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 1.41$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 1.49$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 1.65$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 1.67$

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Answers

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$\copyright \ 2014 \ \ \text{ Dan Ma}$

Exam P Practice Problem 87 – Modeling Insurance Payments

By Dan Ma on July 2, 2014 | Leave a comment

Problem 87-A

A business owner is facing a risk whose economic loss is modeled by the random variable $X$ . The following is the density function of $X$ .

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

The business owner purchases an insurance policy to cover this potential loss. The insurance policy pays the business owner 80% of the amount of each loss.

Given that a loss has occurred, what is the probability that the amount of the insurance payment to the business owner is less than 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.36$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.64$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.75$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.86$

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Problem 87-B

An individual purchases an insurance policy to cover a loss $X$ whose density function is:

$\displaystyle f(x)=\frac{1}{1800} \ x \ \ \ \ \ \ \ \ 0<x<60$

The insurance policy reimburses the policy owner 50% of each loss. Given that a loss has occurred, what is the median amount of the insurance payment made to the policy owner?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 15.00$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 18.65$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 21.21$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 23.63$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 42.43$

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Answers

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$\copyright \ 2014 \ \ \text{ Dan Ma}$

Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, CAS Exam 1, Exam P, Exam P Practice Problems, Insurance and risk management, Median, Probability, SOA Exam P

Exam P Practice Problem 84 – When Random Loss is Doubled

By Dan Ma on June 19, 2014 | Leave a comment

Problem 84-A

A business owner faces a risk whose economic loss amount $X$ follows a uniform distribution over the interval $0<x<1$ . In the next year, the loss amount is expected to be doubled and is expected to be modeled by the random variable $Y=2X$ .

Suppose that the business owner purchases an insurance policy effective at the beginning of next year with the provision that any loss amount less than or equal to 0.5 is the responsibility of the business owner and any loss amount in excess of 0.5 is the responsibility of the insurer. When a loss occurs next year, what is the expected payment made by the insurer to the business owner?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \frac{4}{16}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \frac{6}{16}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \frac{7}{16}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \frac{8}{16}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \frac{9}{16}$

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Problem 84-B

A business owner faces a risk whose economic loss amount $X$ has the following density function:

$\displaystyle f(x)=\frac{x}{2} \ \ \ \ \ \ 0<x<2$

In the next year, the loss amount is expected to be doubled and is expected to be modeled by the random variable $Y=2X$ .

Suppose that the business owner purchases an insurance policy effective at the beginning of next year with the provision that any loss amount less than or equal to 1 is the responsibility of the business owner and any loss amount in excess of 1 is the responsibility of the insurer. When a loss occurs next year, what is the expected payment made by the insurer to the business owner?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{12}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \frac{2}{3}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \frac{19}{12}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \frac{27}{16}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \frac{21}{12}$

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Answers

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$\copyright \ 2014 \ \ \text{ Dan Ma}$

Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, CAS Exam 1, Exam P, Exam P Practice Problems, Expected Insurance Payment, Insurance and risk management, Probability, SOA Exam P, Uniform distribution

Exam P Practice Problem 83 – Claim Size of Auto Insurance Policies

By Dan Ma on June 18, 2014 | Leave a comment

Problem 83-A

An insurance company has a block of auto insurance policies. The claim size (in thousands) for a policy in this block of auto insurance policies is modeled by the random variable $Y=X^2$ where $X$ has a normal distribution with mean 0 and variance 1.5.

What is the expected claim size for such an auto insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 1250$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 1500$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 1750$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 2250$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 2500$

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Problem 83-B

What is the standard deviation of the claim size for such an auto insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 1732$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 3000$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 4243$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 4987$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 5732$

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Answers

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$\copyright \ 2014 \ \ \text{Dan Ma}$

Posted in: Insurance and Risk Management, Normal Distribution, Practice Problems | Tagged: Actuarial Exam, CAS Exam 1, Exam P, Exam P Practice Problems, Insurance and risk management, Normal Distribution, Probability, SOA Exam P, Variance

Exam P Practice Problem 80 – Total Insurance Payment

By Dan Ma on June 15, 2014 | Leave a comment

Problem 80-A

An individual purchases an insurance policy to cover a random loss. If a random loss occurs during the year, the amount of loss is at least 1. Once a random loss occurs, the insurance payment to the insured is modeled by the random variable $X$ with the following density function

$\displaystyle f(x)=\frac{1}{x^2} \ \ \ \ \ 1<x<\infty$

If there is a loss, there is only one loss in each year. In each year, the probability of a loss is 0.25. What is the probability that the annual amount paid to the policyholder under this policy is less than 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.250$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.500$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.750$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.875$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.925$

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Problem 80-B

An individual purchases an insurance policy to cover a random loss. If a random loss occurs during the year, the loss amount is at least 1. Once a loss occurs, the insurance payment to the insured is modeled by the random variable $X$ with the following density function

$\displaystyle f(x)=\frac{1}{30} \ x(1+3x) \ \ \ \ \ 1<x<3$

If there is a loss, there is only one loss in each year. In each year, the probability of a loss is 0.15. What is the probability that the annual amount paid to the policyholder under this policy is less than 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.1500$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.2833$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.8500$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8735$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.8925$

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Answers

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$\copyright \ 2014$

Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, CAS Exam 1, Exam P, Exam P Practice Problems, Insurance and risk management, Probability, SOA Exam P, The law of total probability

Exam P Practice Problem 77 – Estimating Random Claim Sizes

By Dan Ma on May 28, 2013 | Leave a comment

Problem 77-A

The probability distribution of the claim size from an auto insurance policy randomly selected from a large pool of policies is described by the following density function.

$\displaystyle f(x)=\frac{3}{1000} \ (50-5x+\frac{1}{8} \ x^2), \ \ \ \ \ \ \ \ \ \ 0<x<20$

What is the probability that a randomly selected claim from this insurance policy is within 120% of the mean claim size?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.50$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.85$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.88$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.91$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.95$

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Problem 77-B

The probability distribution of the claim size from an auto insurance policy randomly selected from a large pool of policies is described by the following density function.

$\displaystyle f(x)=\frac{3}{2500} \ (100x-20x^2+ x^3), \ \ \ \ \ \ \ \ \ \ 0<x<10$

What is the probability that a randomly selected claim from this insurance policy is within one-half of a standard deviation of the mean claim size?

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.37$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.60$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.62$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.64$

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Answer Keys

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$\copyright \ 2013 \ \ \text{Dan Ma}$

Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, CAS Exam 1, Exam P, Exam P Practice Problems, Expected Value, Insurance and risk management, Probability, SOA Exam P, Standard Deviation, Variance

Exam P Practice Problem 76 – Quantifying Average Random Loss

By Dan Ma on May 27, 2013 | Leave a comment

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?

$\text{ }$

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

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

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

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

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

$\text{ }$

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Problem 76-B

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

$\text{ }$

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

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

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

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

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

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Answer Keys

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$\copyright \ 2013 \ \ \text{Dan Ma}$

Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, Binomial distribution, CAS Exam 1, Exam P, Exam P Practice Problems, Expected Value, Independent Random Varaibles, Independent sum, Insurance and risk management, Probability, SOA Exam P

Exam P Practice Problem 74 – Review of Auto Insurance Claims

By Dan Ma on May 19, 2013 | Leave a comment

Both Problem 74-A and Problem 74-B use the following information.

An insurer issued policies to cover a large number of automobiles. Claim amounts (in thousands) from these policies are independent and are modeled by a continuous uniform distribution on (0,10).

The insurer randomly selects five claims for review.

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Problem 74-A

What is the probability that the minimum claim amount is between 2 thousands and 6 thousands?

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.31$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.32$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.33$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.75$

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Problem 74-B

What is the expected value of the maximum claim amount?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5.0$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5.5$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.6$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.3$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.5$

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Answer Keys

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Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, CAS Exam 1, Exam P, Exam P Practice Problems, Expected Value, Independent Random Varaibles, Insurance and risk management, Order Statistics, Probability, SOA Exam P, Uniform distribution

Exam P Practice Problem 72 – Risk Categories of Insured Drivers

By Dan Ma on May 15, 2013 | Leave a comment

Both Problem 72-A and Problem 72-B use the following information.

A large pool of insured drivers consists of three distinct risk categories – low risk drivers, medium risk drivers and high risk drivers. The following table has more information about these insured drivers.

$\displaystyle \begin{bmatrix} \text{Risk}&\text{ }&\text{ }&\text{Percentage} &\text{ }&\text{ }&\text{Probability of} \\\text{Category}&\text{ }&\text{ }&\text{ } &\text{ }&\text{ }&\text{at Least One Collision} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{Low Risk}&\text{ }&\text{ }&\displaystyle 50 \%&\text{ }&\text{ }&\displaystyle 0.10 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{Medium Risk}&\text{ }&\text{ }&30 \%&\text{ }&\text{ }&0.20 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ \text{High Risk}&\text{ }&\text{ }&20 \%&\text{ }&\text{ }&0.50 \end{bmatrix}$

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Problem 72-A

Three insured drivers are randomly selected from this large pool of insured drivers.

What is the probability that all three insured drivers are drawn from different risk categories?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.01$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.03$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.06$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.18$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.36$

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Problem 72-B

Four insured drivers are randomly selected from this large pool of insured drivers.

What is the probability that all three risk categories are represented in these four insured drivers?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.096$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.072$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.108$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.180$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.360$

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Answer Keys

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Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, CAS Exam 1, Exam P, Exam P Practice Problems, Insurance and risk management, Multinomial Distribution, Probability, SOA Exam P

Exam P Practice Problem 71 – Estimating Claim Frequency

By Dan Ma on May 1, 2013 | 1 Comment

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?

$\text{ }$

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

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

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

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

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

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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?

$\text{ }$

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

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

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

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

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

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Answer Keys

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Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, Bayes' Theorem, Binomial distribution, CAS Exam 1, Exam P, Exam P Practice Problems, Geometric distribution, Insurance and risk management, Probability, SOA Exam P, Variance

SOA Exam P / CAS Exam 1

Category Archives: Insurance and Risk Management

Exam P Practice Problem 88 – Expected Value of Insurance Payments

Exam P Practice Problem 87 – Modeling Insurance Payments

Exam P Practice Problem 84 – When Random Loss is Doubled

Exam P Practice Problem 83 – Claim Size of Auto Insurance Policies

Exam P Practice Problem 80 – Total Insurance Payment

Exam P Practice Problem 77 – Estimating Random Claim Sizes

Exam P Practice Problem 76 – Quantifying Average Random Loss

Exam P Practice Problem 74 – Review of Auto Insurance Claims

Exam P Practice Problem 72 – Risk Categories of Insured Drivers

Exam P Practice Problem 71 – Estimating Claim Frequency

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