Monthly Archives: May, 2013

Exam P Practice Problem 78 – Tracking A Group of Insureds

Problem 78-A

An insurance company tracked a group of 625 insureds for 2 years. It was found that 370 of the insureds had no claims in year 1 and 395 of the insureds had no claims in year 2. Furthermore, there were five times as many insureds who had no claims in both years than there were insureds who had claims in both years.

Select an insured at random from this group. What is the probability that the randomly selected insured had no claims in both years?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.23

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.28

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.44

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.47

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Problem 78-B

An insurance company tracked a group of 1000 insureds for 2 years. It was found that 550 of the insureds had no claims in year 1 and 680 of the insureds had at least one claim in year 2. Of the insureds who had at least one claim in year 1, 306 insureds had at least one claim in year 2.

Select an insured at random from this group. Given that this insured had no claims in year 1, what is the probability that the selected insured had no claims in year 2?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.176

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.230

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.320

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.418

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.764

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

Exam P Practice Problem 77 – Estimating Random Claim Sizes

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?

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

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

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

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

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

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

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

Exam P Practice Problem 75 – Travel Time to Work By Train

Both Problem 75-A and Problem 75-B use the following information.

A worker travels to work by train 5 days a week (Monday to Friday). The length of a train ride (in minutes) to work follows a continuous uniform distribution from 10 to 40.

The lengths of the train ride across the days of the week are independent.

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Problem 75-A

What is the probability that the shortest train ride during a work week is between 15 and 20 minutes?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.025

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.039

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.045

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.053

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.064

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Problem 75-B

What is the expected value of the longest train ride during a work week?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 25.0

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 25.9

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 28.2

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 33.3

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 35.7

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\copyright \ 2013-2016 \ \ \text{Dan Ma}

Exam P Practice Problem 74 – Review of Auto Insurance Claims

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?

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

Exam P Practice Problem 73 – Wait Time at a Busy Restaurant

Both Problem 73-A and Problem 73-B use the following information.

A certain restaurant is very busy in the evening time during the weekend. When customers arrive, they typically have to wait for a table.

When a customer has to wait for a table, the wait time (in minutes) follows a distribution with the following density function.

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

A customer plans to dine at this restaurant on five Saturday evenings during the next 3 months. Assume that the customer will have to wait for a table on each of these evenings.

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Problem 73-A

What is the probability that the minimum wait time for a table during the next 3 months for this customer will be more than half an hour?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.24

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.36

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.42

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

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Problem 73-B

What is the mean of the maximum wait time (in minutes) for a table during the next 3 months for this customer?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 40.0

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 50.5

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 51.4

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 54.5

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 55.4

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

Exam P Practice Problem 72 – Risk Categories of Insured Drivers

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

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