Exam P Practice Problem 70 – Real Estate Sales Contest

Problem 70-A

A commercial real estate property company has three sales agents who are actively selling commercial real estate properties. The times (in days) to the next successful sale for these three agents are exponentially distributed with means 10 days, 15 days and 20 days.

These three agents work independently. So the time to the next successful sale for one agent is independent of the time to the next successful sale for any of the other agents.

To spur sales, the company has a contest among the three agents. Each agent produces a sale. The award will go to the first agent producing the first sale.

What is the probability that the winning sale will take place within one week?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.14$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.22$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.50$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.78$

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

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Problem 70-B

A commercial real estate property company has four sales agents who are actively selling commercial real estate properties. The times (in days) to the next successful sale for these four agents are exponentially distributed with means 10 days, 15 days and 20 days and 30 days.

These four agents work independently. So the time to the next successful sale for one agent is independent of the time to the next successful sale for any of the other agents.

To spur sales, the company has a contest among the four agents. Each agent produces a sale. The award will go to the first agent producing the first sale.

What is the expected waiting time (in days) from the beginning of the contest to the occurrence of the winning sale?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10$

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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 68 – Large Claim Studies

Problem 68-A

An insurance company has a large block of auto insurance policies such that the claim sizes (in thousands) from a policy in this block are modeled by the random variable $X$. The following is the probability density function of $X$.

$\displaystyle f(x)=\frac{3}{16000} \ (400-x^2) \ \ \ \ \ \ \ 0

An actuary is hired to study the large claims arising from these auto insurance policies. In this study, any claim size over ten thousands is considered a large claim.

What is the mean size of the claims studied by this actuary?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7,500$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 13,500$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 14,219$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15,000$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 17,500$

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Problem 68-B

An insurance company has a large block of auto insurance policies such that the claim sizes (in thousands) from a policy in this block are modeled by the random variable $X$. The following is the probability density function of $X$.

$\displaystyle f(x)=\frac{625}{312 \ x^3} \ \ \ \ \ \ \ 1

An actuary is hired to study the large claims arising from these auto insurance policies. In this study, any claim size over five thousands is considered a large claim.

What is the mean size of the claims studied by this actuary?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1,923$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6,923$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5,321$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8,333$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15,000$

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Exam P Practice Problem 67 – Statistical Studies of Insured Drivers

Problem 67-A

An auto insurance company performed a statistical study on its insured drivers. The following table shows the results.

$\displaystyle \begin{bmatrix} \text{Age Group}&\text{ }&\text{Percentage}&\text{ }&\text{Annual Probability of} \\ \text{ }&\text{ }&\text{of its Drivers}&\text{ }&\text{At Least One Claim} \\\text{ }&\text{ }&\text{ } \\ \text{16-20}&\text{ }&15 \% &\text{ }&0.18 \\\text{ }&\text{ }&\text{ } \\ \text{21-30}&\text{ }&20 \% &\text{ }&0.12 \\\text{ }&\text{ }&\text{ } \\ \text{31-50}&\text{ } &30 \% &\text{ }&0.08 \\\text{ }&\text{ }&\text{ } \\ \text{51-70}&\text{ }&25 \% &\text{ }&0.09 \\\text{ }&\text{ }&\text{ } \\ \text{71 and up}&\text{ }&10 \% &\text{ }&0.11 \end{bmatrix}$

The authors of the statistical study also found that for any insured driver in the study, the annual number of claims follows a Poisson distribution.

Suppose that an insured driver in the study had exactly 2 claims in the past year. What is the probability that the insured driver is from the age group 16-20?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.150$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.223$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.249$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.376$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.415$

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Problem 67-B

An auto insurance company performed a statistical study on its younger insured drivers (under 35 years of age). The following table shows the results.

$\displaystyle \begin{bmatrix} \text{Age Group}&\text{ }&\text{Percentage}&\text{ }&\text{Annual Probability of} \\ \text{ }&\text{ }&\text{of its Drivers}&\text{ }&\text{At Least One Claim} \\\text{ }&\text{ }&\text{ } \\ \text{16-17}&\text{ }&12 \% &\text{ }&0.18 \\\text{ }&\text{ }&\text{ } \\ \text{18-24}&\text{ }&38 \% &\text{ }&0.10 \\\text{ }&\text{ }&\text{ } \\ \text{25-34}&\text{ } &50 \% &\text{ }&0.06 \end{bmatrix}$

The authors of the statistical study also found that for any insured driver in the study, the annual number of claims follows a Poisson distribution. Furthermore, for any insured driver in the study, the number of claims in one year is independent of the number of claims in any other year.

Suppose that in a 2-year period, an insured driver in the study had exactly 1 claim in year 1 and exactly 2 claims in year 2. What is the probability that the insured driver is from the age group 16-17?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.120$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.229$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.241$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.329$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.576$

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Exam P Practice Problem 66 – Median Cholesterol Level

Problem 66-A

The blood cholesterol levels of men aged 55 to 64 are normally distributed with mean 225 milligrams per deciliter (mg/dL) and standard deviation 39.5 mg/dL.

A medical researcher is planning a clinical study targeting men from the age group of 55 to 64 who have high blood cholesterol levels (above 240 mg/dL).

What is the median cholesterol level of the men in the target population of this medical study?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 225 \ \text{ mg/dL}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 249 \ \text{ mg/dL}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 262 \ \text{ mg/dL}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 284 \ \text{ mg/dL}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 294 \ \text{ mg/dL}$

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Problem 66-B

The blood cholesterol levels of women aged 55 to 64 are normally distributed with mean 190 milligrams per deciliter (mg/dL) and standard deviation 40 mg/dL.

A medical researcher is planning a clinical study targeting women from the age group of 55 to 64 who have borderline high blood cholesterol levels (between 200 and 240 mg/dL).

What is the median cholesterol level of the women in the target population of this medical study?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 214 \ \text{ mg/dL}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 216 \ \text{ mg/dL}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 218 \ \text{ mg/dL}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 220 \ \text{ mg/dL}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 225 \ \text{ mg/dL}$

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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 64 – Median Claim Size

Problem 64-A

The following is the density function of the claim size $X$ of an insurance policy from a block of property and casualty insurance policies:

$\displaystyle f(x)=\frac{1}{2} \biggl( 0.4 \ e^{-0.4 x} \biggr) + \frac{1}{2} \biggl( 0.2 \ e^{-0.2 x} \biggr) \ \ \ \ \ \ \ 0

Calculate the median claim size $X$.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.618$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.739$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.406$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.599$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.533$

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Problem 64-B

The following is the density function of the claim size $X$ of an insurance policy from a block of auto insurance policies:

$\displaystyle f(x)=\frac{3}{4} \biggl( \frac{1}{8} \ x \biggr) + \frac{1}{4} \biggl( \frac{1}{64} \ x^3 \biggr) \ \ \ \ \ \ \ 0

Calculate the median claim size $X$.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.2216$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.4972$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.8653$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.9622$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.9975$

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Exam P Practice Problem 63 – Total Minutes of Telephone Calls

Problem 63-A

For a certain individual, the daily number of telephone calls (incoming or outgoing) has a Poisson distribution with mean 12. The length in time (in minutes) of each telephone call has an exponential distribution with mean 5 minutes.

The length of time of one telephone call is independent of the length of time of any other telephone call.

On a given day, this individual makes or receives 4 telephone calls. What is the probability that this person is on the telephone for more than half an hour?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1218$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1260$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1456$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1490$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1512$

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Problem 63-B

For a certain individual, the daily number of telephone calls (incoming or outgoing) has a Poisson distribution with mean 16. The length in time (in minutes) of each telephone call has an exponential distribution with mean 8 minutes.

The length of time of one telephone call is independent of the length of time of any other telephone call.

On a given day, this individual makes or receives 5 telephone calls. What is the probability that this person is on the telephone for more than 45 minutes?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2237$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2596$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3384$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3975$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.4085$

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Exam P Practice Problem 62 – Waiting for Telephone Calls

Problem 62-A

An individual classifies the telephone calls he receives into the categories of Personal Calls (e.g. calls from friends and relatives) and Non-Personal Calls (all the other calls that are considered not Personal Calls).

Let $X$ be the time (in minutes) until the next Personal Call. Let $Y$ be the time (in minutes) until the next Non-Personal Call.

Suppose that $X$ and $Y$ are independent random variables and follow exponential distributions with means 8 minutes and 3 minutes, respectively.

What is the probability that the next incoming telephone call is a Personal Call?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2727$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3735$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.5000$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6265$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.7273$

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Problem 62-B

An individual classifies the telephone calls he receives into the categories of Personal Calls (e.g. calls from friends and relatives), Business Calls (calls related to his small business) and Other Calls (all the other calls not belonging to the Personal Call and Business Call categories).

Let $X$ be the time (in minutes) until the next Personal Call, let $Y$ be the time (in minutes) until the next Business Call and let $Z$ be the time (in minutes) until the next Other Call.

Suppose $X$, $Y$ and $Z$ are independent random variables and follow exponential distributions with means 12, 10 and 6 minutes, respectively.

What is the probability that the next telephone call this individual receives will be a Business Call?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{3}{14}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{4}{14}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{3}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{14}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{6}{14}$

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