# Exam P Practice Problem 110 – likelihood of auto accidents

Problem 110-A

An actuary studied the likelihood of accidents in a one-year period among a large group of insured drivers. The following table gives the results.

Age Group Percent of Drivers Probability of 0 Accidents Probability of 1 Accident
16-20 15% 0.20 0.25
21-30 25% 0.35 0.40
31-50 35% 0.60 0.30
51-70 20% 0.67 0.23
71+ 5% 0.50 0.35

Suppose that a randomly selected insured driver in the studied group had at least 2 accidents in the past year. Calculate the probability that the insured driver is in the age group 21-30.

$\displaystyle \bold ( \bold A \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 1 \bold 7$

$\displaystyle \bold ( \bold B \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 2 \bold 4$

$\displaystyle \bold ( \bold C \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 3 \bold 0$

$\displaystyle \bold ( \bold D \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 4 \bold 0$

$\displaystyle \bold ( \bold E \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 4 \bold 5$

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Problem 110-B

An auto insurance company performed a study on the frequency of accidents of its insured drivers in a one-year period. The following table gives the results of the study.

Age Group Percent of Drivers Probability of At Least 1 Accident
16-20 10% 0.30
21-40 20% 0.20
41-65 35% 0.10
66+ 35% 0.12

A randomly selected insured driver from the study was found to have no accidents in the one-year period.

Calculate the probability that the insured driver is from the age group 16-20.

$\displaystyle \bold ( \bold A \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 0 \bold 8$

$\displaystyle \bold ( \bold B \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 1 \bold 2$

$\displaystyle \bold ( \bold C \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 1 \bold 5$

$\displaystyle \bold ( \bold D \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 1 \bold 9$

$\displaystyle \bold ( \bold E \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 2 \bold 0$

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# Exam P Practice Problem 109 – counting insurance payments

Problem 109-A

Amounts of damages due to auto collision accidents follow a probability distribution whose density function is given by the following.

$\displaystyle f(x) = \left\{ \begin{array}{ll} \displaystyle \frac{3}{8000} \ (400-40x+x^2) &\ \ \ \ \ \ 0 < x < 20 \\ \text{ } & \text{ } \\ \displaystyle 0 &\ \ \ \ \ \ \text{otherwise} \\ \end{array} \right.$

When occurred, the collision damages are reimbursed by an insurance coverage subject to a deductible of 4.

Fifteen unrelated auto collision accidents have been reported. Determine the probability that exactly nine or ten of the accidents will be reimbursed by the insurance coverage.

$\displaystyle \bold ( \bold A \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 1 \bold 4 \bold 2$

$\displaystyle \bold ( \bold B \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 1 \bold 6 \bold 3$

$\displaystyle \bold ( \bold C \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 2 \bold 2 \bold 2$

$\displaystyle \bold ( \bold D \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 2 \bold 6 \bold 6$

$\displaystyle \bold ( \bold E \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 2 \bold 8 \bold 9$

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Problem 109-B

Amounts of damages due to auto collision accidents follow a probability distribution whose density function is given by the following.

$\displaystyle f(x) = \left\{ \begin{array}{ll} \displaystyle \frac{3}{4000} \ (400-80x+4 x^2) &\ \ \ \ \ \ 0 < x < 10 \\ \text{ } & \text{ } \\ \displaystyle 0 &\ \ \ \ \ \ \text{otherwise} \\ \end{array} \right.$

When occurred, the damages are reimbursed by an insurance coverage subject to a deductible of 2.

Twelve unrelated auto collision accidents have been reported. Determine the probability that exactly six or seven of the accidents will not be reimbursed by the insurance coverage.

$\displaystyle \bold ( \bold A \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 1 \bold 8 \bold 4$

$\displaystyle \bold ( \bold B \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 2 \bold 2 \bold 5$

$\displaystyle \bold ( \bold C \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 4 \bold 0 \bold 8$

$\displaystyle \bold ( \bold D \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 4 \bold 2 \bold 7$

$\displaystyle \bold ( \bold E \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 0 \bold . \bold 4 \bold 5 \bold 0$

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# Exam P Practice Problem 102 – estimating claim costs

Problem 102-A

Insurance claims modeled by a distribution with the following cumulative distribution function.

$\displaystyle F(x) = \left\{ \begin{array}{ll} \displaystyle 0 &\ \ \ \ \ \ x \le 0 \\ \text{ } & \text{ } \\ \displaystyle \frac{1}{1536} \ x^4 &\ \ \ \ \ \ 0 < x \le 4 \\ \text{ } & \text{ } \\ \displaystyle 1-\frac{2}{3} x+\frac{1}{8} x^2- \frac{1}{1536} \ x^4 &\ \ \ \ \ \ 4 < x \le 8 \\ \text{ } & \text{ } \\ \displaystyle 1 &\ \ \ \ \ \ x > 8 \\ \end{array} \right.$

The insurance company is performing a study on all claims that exceed 3. Determine the mean of all claims being studied.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 4.8$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 4.9$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 5.0$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 5.1$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 5.2$

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Problem 102-B

Insurance claims modeled by a distribution with the following cumulative distribution function.

$\displaystyle F(x) = \left\{ \begin{array}{ll} \displaystyle 0 &\ \ \ \ \ \ x \le 0 \\ \text{ } & \text{ } \\ \displaystyle \frac{1}{50} \ x^2 &\ \ \ \ \ \ 0 < x \le 5 \\ \text{ } & \text{ } \\ \displaystyle -\frac{1}{50} x^2+\frac{2}{5} x- 1 &\ \ \ \ \ \ 5 < x \le 10 \\ \text{ } & \text{ } \\ \displaystyle 1 &\ \ \ \ \ \ x > 10 \\ \end{array} \right.$

The insurance company is performing a study on all claims that exceed 4. Determine the mean of all claims being studied.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 5.9$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 6.0$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 6.1$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 6.2$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 6.3$

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# Exam P Practice Problem 100 – find the variance of loss in profit

Problem 100-A

The monthly amount of time $X$ (in hours) during which a manufacturing plant is inoperative due to equipment failures or power outage follows approximately a distribution with the following moment generating function.

$\displaystyle M(t)=\biggl( \frac{1}{1-7.5 \ t} \biggr)^2$

The amount of loss in profit due to the plant being inoperative is given by $Y=12 X + 1.25 X^2$.

Determine the variance of the loss in profit.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \text{279,927.20}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \text{279,608.20}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \text{475,693.76}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \text{583,358.20}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \text{601,769.56}$

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Problem 100-B

The weekly amount of time $X$ (in hours) that a manufacturing plant is down (due to maintenance or repairs) has an exponential distribution with mean 8.5 hours.

The cost of the downtime, due to lost production and maintenance and repair costs, is modeled by $Y=15+5 X+1.2 X^2$.

Determine the variance of the cost of the downtime.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \text{130,928.05}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \text{149,368.45}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \text{181,622.05}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \text{188,637.67}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \text{195,369.15}$

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# Exam P Practice Problem 99 – When Random Loss is Doubled

Problem 99-A

A business owner faces a risk whose economic loss amount $X$ follows a uniform distribution over the interval $0. 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 that is greater than 0.5 is paid by the insurer in full. When a loss occurs next year, determine the expected payment made by the insurer to the business owner.

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

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

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \frac{13}{16}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \frac{15}{16}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \frac{17}{16}$

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Problem 99-B

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

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

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 that is greater than 1 is paid by the insurer in full. When a loss occurs next year, what is the expected payment made by the insurer to the business owner?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.6667$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 1.5833$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 1.6875$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 1.7500$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 2.6250$

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# Exam P Practice Problem 97 – Variance of Claim Sizes

Problem 97-A

For a type of insurance policies, the following is the probability that the size of claim is greater than $x$.

$\displaystyle P(X>x) = \left\{ \begin{array}{ll} \displaystyle 1 &\ \ \ \ \ \ x \le 0 \\ \text{ } & \text{ } \\ \displaystyle \biggl(1-\frac{x}{10} \biggr)^6 &\ \ \ \ \ \ 0

Calculate the variance of the claim size for this type of insurance policies.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \frac{10}{7}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \frac{75}{49}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \frac{95}{49}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \frac{15}{7}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \frac{25}{7}$

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Problem 97-B

For a type of insurance policies, the following is the probability that the size of a claim is greater than $x$.

$\displaystyle P(X>x) = \left\{ \begin{array}{ll} \displaystyle 1 &\ \ \ \ \ \ x \le 0 \\ \text{ } & \text{ } \\ \displaystyle \biggl(\frac{250}{x+250} \biggr)^{2.25} &\ \ \ \ \ \ x>0 \\ \end{array} \right.$

Calculate the expected claim size for this type of insurance policies.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 200.00$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 203.75$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 207.67$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 217.32$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 232.74$

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# Exam P Practice Problem 96 – Expected Insurance Payment

Problem 96-A

An insurance policy is purchased to cover a random loss subject to a deductible of 1. The cumulative distribution function of the loss amount $X$ is:

$\displaystyle F(x) = \left\{ \begin{array}{ll} \displaystyle 0 &\ \ \ \ \ \ x<0 \\ \text{ } & \text{ } \\ \displaystyle \frac{3}{25} \ x^2 - \frac{2}{125} \ x^3 &\ \ \ \ \ \ 0 \le x<5 \\ \text{ } & \text{ } \\ 1 &\ \ \ \ \ \ 5

Given a random loss $X$, determine the expected payment made under this insurance policy.

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

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 1.54$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 1.72$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 4.63$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 6.26$

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Problem 96-B

An insurance policy is purchased to cover a random loss subject to a deductible of 2. The density function of the loss amount $X$ is:

$\displaystyle f(x) = \left\{ \begin{array}{ll} \displaystyle \frac{3}{8} \biggl(1- \frac{1}{4} \ x + \frac{1}{64} \ x^2 \biggr) &\ \ \ \ \ \ 0

Given a random loss $X$, what is the expected benefit paid by this insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.51$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.57$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.63$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 1.60$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 2.00$

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

# Exam P Practice Problem 93 – Determining Average Claim Frequency

Problem 93-A

An actuary performs a claim frequency study on a group of auto insurance policies. She finds that the probability function of the number of claims per week arising from this set of policies is $P(N=n)$ where $n=1,2,3,\cdots$. Furthermore, she finds that $P(N=n)$ is proportional to the following function:

$\displaystyle \frac{e^{-2.9} \cdot 2.9^n}{n!} \ \ \ \ \ \ \ n=1,2,3,\cdots$

What is the weekly average number of claims arising from this group of insurance policies?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 2.900$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 3.015$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 3.036$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 3.069$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 3.195$

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Problem 93-B

Let $N$ be the number of taxis arriving at an airport terminal per minute. It is observed that there are at least 2 arrivals of taxis in each minute. Based on a study performed by a traffic engineer, the probability $P(N=n)$ is proportional to the following function:

$\displaystyle \frac{e^{-2.9} \cdot 2.9^n}{n!} \ \ \ \ \ \ \ n=2,3,4,\cdots$

What is the average number of taxis arriving at this airport terminal per minute?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 2.740$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 2.900$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 3.339$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 3.489$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 3.692$

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$\copyright \ 2015 \ \ \text{ Dan Ma}$

# Exam P Practice Problem 92 – Expected Claim Payment

Problem 92-A

The size of a claim that an auto insurance company pays out is modeled by a random variable with the following density function:

$\displaystyle f(x)=\frac{1}{5000} \ (100-x) \ \ \ \ \ \ \ 0

By subjecting the insured to a deductible of 12 per claim, what is the expected reduction in claim payment?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 9.50$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 10.6$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 11.1$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 11.8$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 12.0$

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Problem 92-B

The size of a claim that an auto insurance company pays out is modeled by a random variable with the following density function:

$\displaystyle f(x)=\frac{1}{3200} \ (80-x) \ \ \ \ \ \ \ 0

By subjecting the insured to a deductible of 10 per claim, by what percent is the expected claim payment reduced?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 10 \%$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 15 \%$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 22 \%$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 25 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 33 \%$

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$\copyright \ 2015 \ \ \text{ Dan Ma}$

# Exam P Practice Problem 90 – Insurance Benefits

Problem 90-A

A random loss follows an exponential distribution with mean 20. An insurance reimburses this random loss up to a benefit limit of 30.

When a loss occurs, what is the expected value of the benefit not paid by this insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 4.5$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 5.1$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 6.3$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 8.5$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 11.2$

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Problem 90-B

A random loss follows an exponential distribution with mean 100. An insurance reimburses this random loss up to a benefit limit of 200.

When a loss occurs, what is the expected value of the benefit not paid by this insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 12.6$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 13.5$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 24.6$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 40.6$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 40.7$

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