# Exam P Practice Problem 107 – wait time at a busy restaurant

Both 107-A and 107-B use the following probability density function.

$\displaystyle f(x) = \left\{ \begin{array}{ll} \displaystyle \frac{1}{450} \ (30-x) &\ \ \ \ \ \ 0 < x < 30 \\ \text{ } & \text{ } \\ \displaystyle 0 &\ \ \ \ \ \ \text{otherwise} \\ \end{array} \right.$

Problem 107-A

The wait time (in minutes) for a table at a busy restaurant on the weekend is distributed according to the density function $f(x)$ given above.

A customer plans to dine in this restaurant on two different weekends.

Determine the expected value of the longest wait of these two visits to the restaurant.

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

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

$\displaystyle \bold ( \bold C \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 1 \bold 2 \bold . \bold 8$

$\displaystyle \bold ( \bold D \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 1 \bold 3 \bold . \bold 5$

$\displaystyle \bold ( \bold E \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 1 \bold 4 \bold . \bold 0$

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Problem 107-B

The wait time (in minutes) for a table at a busy restaurant on the weekend is distributed according to the density function $f(x)$ given above.

A customer plans to dine in this restaurant on two different weekends.

Determine the expected value of the shortest wait of these two visits to the restaurant.

$\displaystyle \bold ( \bold A \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 4 \bold . \bold 5$

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

$\displaystyle \bold ( \bold C \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 7 \bold . \bold 0$

$\displaystyle \bold ( \bold D \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 8 \bold . \bold 6$

$\displaystyle \bold ( \bold E \bold ) \ \ \ \ \ \ \ \ \ \ \ \ \bold 1 \bold 0 \bold . \bold 0$

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# 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 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

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 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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$\copyright \ 2013$

# Exam P Practice Problem 45 – Heights of Male Students

Problem 45-A

Heights of male students in a large university follow a normal distribution with mean 69 inches and standard deviation 2.8 inches.

Five male students from this university are randomly selected.

What is the probability that the shortest student among the five randomly selected students is taller than 5 feet 5 inches?

Note that one feet = 12 inches.

$\displaystyle A. \ \ \ \ \ \ \ \ \ \ (0.0764)^5$

$\displaystyle B. \ \ \ \ \ \ \ \ \ \ 0.0764$

$\displaystyle C. \ \ \ \ \ \ \ \ \ \ 0.3279$

$\displaystyle D. \ \ \ \ \ \ \ \ \ \ 0.6721$

$\displaystyle E. \ \ \ \ \ \ \ \ \ \ 0.9236$

The answers are based on this normal table from SOA.

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Problem 45-B

Heights of male students in a large university follow a normal distribution with mean 69 inches and standard deviation 2.8 inches.

Ten male students from this university are randomly selected.

What is the probability that the tallest student among the ten randomly selected students is shorter than 6 feet 2 inches?

Note that one feet = 12 inches.

$\displaystyle A. \ \ \ \ \ \ \ \ \ \ (0.0367)^{10}$

$\displaystyle B. \ \ \ \ \ \ \ \ \ \ 0.0367$

$\displaystyle C. \ \ \ \ \ \ \ \ \ \ 0.3120$

$\displaystyle D. \ \ \ \ \ \ \ \ \ \ 0.6880$

$\displaystyle E. \ \ \ \ \ \ \ \ \ \ 0.9633$

The answers are based on this normal table from SOA.

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The answers are based on this normal table from SOA.

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$\copyright \ 2013$

# Exam P Practice Problem 42 – Losses due to Collisions

Problem 42-A

Amount of damage to cars (in thousands) due to collision accidents is distributed according to the following density function:

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

Five independent collision accidents are randomly selected. What is the probability that the maximum amount of damage from these five accidents is greater than 2?

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Problem 42-B

A certain type of random losses is distributed according to the following density function:

$\displaystyle f(x)=\frac{1}{18} \ x \ \ \ \ \ \ 0

Three independent loss amounts are randomly selected. What is the probility that the minimum amount of these losses is no greater than 3?

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$\copyright \ 2013$