# Exam P Practice Problem 85 – Supplemental Coverages for Employees

Problem 85-A

A large employer offers three supplemental benefits in addition to the major medical benefit – dental, vision and group life insurance – to its employees. The following information is known.

• Thirty five percent of the employees choose dental and vision.
• Twelve percent of the employees choose all three supplemental benefits.
• Fifty six percent of the employees choose exactly two of the three supplemental benefits.
• Sixteen percent of the employees choose neither vision nor dental.
• Five percent of the employees choose none of the three supplemental benefits.

One employee is chosen at random. What is the probability that this employee has group life insurance benefit?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.32$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.56$

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

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.65$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.68$

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Problem 85-B

A professor tracks a group of students taking astrophysics, biostatistics and chemistry. The following information is known.

• Forty five percent of the students pass astrophysics and biostatistics.
• Nineteen percent of the students pass biostatistics and chemistry only.
• Twenty two percent of the students pass neither biostatistics nor chemistry.
• Sixty seven percent of the students pass biostatistics.
• Thirty four percent of the students pass astrophysics and biostatistics only.

One student is chosen at random. What is the probability that this student passes chemistry?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.35$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.40$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.41$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.47$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.53$

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

# Exam P Practice Problem 84 – When Random Loss is Doubled

Problem 84-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 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

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

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

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

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 3.

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

# Exam P Practice Problem 82 – Estimating the Median Weight of Bears

Problem 82-A

A wildlife biologist wished to estimate the median weight of bears in Alaska. The weights of the bear population he studied follow a continuous distribution with an unknown median $M$. He captured a sample of 15 bears. Let $Y_5$ be the weight (in pounds) of the fifth smallest bear in the sample of 15 captured bears. Let $Y_{11}$ be the weight (in pounds) of the fifth largest bear in the sample.

Calculate the probability that the median $M$ is between $Y_5$ and $Y_{11}$, i.e., $P(Y_5.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.5000$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.7899$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.8218$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8815$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.9232$

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Problem 82-B

The wildlife biologist in Problem 82-A also wishes to estimate $\tau_{75}$, the seventy fifth percentile of the weights of bear population he studied. Let $Y_{10}$ be the weight of the tenth smallest bear in the sample of 15 captured bears. Let $Y_{14}$ be the weight of the second largest bear in the sample of 15 bears.

Calculate the probability that $\tau_{75}$ is between $Y_{10}$ and $Y_{14}$, i.e., $P(Y_{10}<\tau_{75}.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.6155$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.7500$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.7715$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8383$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.9232$

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

# Exam P Practice Problem 81 – A Medical Survey

Problem 81-A

A medical researcher tracked a group of 10,000 patients with heart disease and diabetes. The researcher found that sixty five percent of these patients have health insurance coverage. She also found that thirty percent and twenty percent of the patients with health insurance coverage have heart disease and diabetes, respectively. Of the patients with health insurance coverage, ten percent have both heart disease and diabetes. Furthermore, the medical researcher found that forty percent and twenty five percent of the patients with no health insurance coverage have heart disease and diabetes, respectively. Twenty percent of the patients with no health insurance coverage have both heart disease and diabetes.

What is the number of patients that have neither heart disease nor diabetes?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 1500$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 2600$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 4175$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 5825$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 8500$

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Problem 81-B

A large high school offers three sporting activities – swimming, running and basketball. Thirty percent of the students participate in swimming. Of the students participating in swimming, fifty percent participate in running, thirty percent participate in basketball and five percent participate in both running and basketball. Of the students not participating in swimming, twenty percent participate in running, forty percent participate in basketball and ten percent participate in both running and basketball.

What is the percentage of the students that participate in exactly two of the sports?

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

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 22.5 \%$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 28 \%$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 35 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 57.5 \%$

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$\copyright \ 2014$

# Exam P Practice Problem 80 – Total Insurance Payment

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

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

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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$\copyright \ 2014$

# Exam P Practice Problem 79 – wait time at a bank

When a customer walks into a branch office of a certain large national bank for routine banking service (e.g. making cash deposits or making cash withdrawals), he or she will be served right away if one of the bank tellers is available. If all bank tellers are busy, the wait time for a bank teller (in minutes) follows a distribution with the following density function:

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

Problem 79-A

On one particular day, the branch office is short staffed. As a result, the wait time for a bank teller is 25% longer than usual. When a customer has to wait, what is the mean wait time on this day?

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

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{20}{12} \text{ minutes}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{25}{12} \text{ minutes}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{40}{12} \text{ minutes}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{50}{12} \text{ minutes}$

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Problem 79-B

On one particular day, the branch office is short staffed. As a result, the wait time for an bank teller is 25% longer than usual. When there is a need to wait on this day, what is the probability that a customer has to wait more than 3.75 minutes for a bank teller?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0625$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0695$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1200$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1600$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1950$

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