# 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 44 – Traffic Statistics Using Poisson Distribution

Problem 44-A

The number of car accidents in a stretch of a highway (Highway #1) has a Poisson distribution with a mean of 4 per week. The number of car accidents in a stretch of another highway (Highway #2) has a Poisson distribution with a mean of 8 per week.

Assume that on a weekly basis, the number of accidents in one highway is independent of the number of accidents in the other highway.

In one particular week, exactly 5 auto accidents took place in these two highways. What is the probability that Highway #1 had exactly 2 accidents in this particular week?

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Problem 44-B

The number of cars (in a day) that break down in a stretch of a highway (Highway #1) has a Poisson distribution with a mean of 16. The number of cars (in a day) that break down in a stretch of another highway (Highway #2) has a Poisson distribution with a mean of 8.

Assume that on a daily basis, the number of cars breaking down in one highway is independent of the number of cars breaking down in the other highway.

In one particular day, exactly 8 cars were found to break down in these two highways. What is the probability that Highway #1 had exactly 5 cars breaking down in this particular day?

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$\copyright \ 2013$

# Exam P Practice Problem 43 – Joint Random Losses

Problem 43-A

Two random losses $X$ and $Y$ are jointly distributed according to the following density function:

$\displaystyle f(x,y)=\frac{1}{64} \ x \ y \ \ \ \ \ \ 0

Suppose that these two random losses had occurred. If the total loss is 5, what is the expected value of the loss $X$?

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Problem 43-B

Two random losses $X$ and $Y$ are jointly distributed according to the following density function:

$\displaystyle f(x,y)=\frac{1}{64} \ (4-x) \ (4-y) \ \ \ \ \ \ 0

Suppose that these two random losses had occurred. If the total loss is 6, what is the expected value of the loss $X$?

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$\copyright \ 2013$

# Exam P Practice Problem 41 – Conditional Expected Number of Balls

Problem 41-A

An insurer has a block of business where the number of claims in a year for a policyholder in the block has the following probability distribution.

$\displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Probability} \\\text{ }&\text{ }&\text{ } \\ 0&\text{ }&\displaystyle 0.08 \\\text{ }&\text{ }&\text{ } \\ 1&\text{ }&\displaystyle 0.35 \\\text{ }&\text{ }&\text{ } \\ 2&\text{ }&\displaystyle 0.40 \\\text{ }&\text{ }&\text{ } \\ 3&\text{ }&\displaystyle 0.12 \\\text{ }&\text{ }&\text{ } \\ 4&\text{ }&\displaystyle 0.0375 \\\text{ }&\text{ }&\text{ } \\ 5&\text{ }&\displaystyle 0.0125 \end{bmatrix}$

Two policyholders from this block are randomly selected and observed for a year. It is found that there are exactly four claims for these two policyholders in the past year. What is the probability that one of the policyholders has exactly three claims?

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Problem 41-B

There are two bowls containing red balls and white balls. Bowl 1 contains 5 red balls and 5 white balls. Five balls are randomly selected from Bowl 1 without replacement. Bowl 2 also contains 5 red balls and 5 white balls. Five balls are randomly selected from Bowl 2 with replacement.

If the total number of red balls drawn from the two bowls is eight, what is the expected number of red balls from Bowl 2?

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$\copyright \ 2013$

# Exam P Practice Problem 37 – Number of Claims

Problem 37A

An insurer issued insurance policies to two independent insureds (Insured #1 and Insured #2). The number of claims in the upcoming year for each insured has a Poisson distribution with mean 1.

1. What is the probability that each of the insureds has at least one claim in the upcoming year?
2. What is the probability that each of the insureds has at least one claim given that there is at least one claim in the upcoming year among these two insureds?

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Problem 37B

The number of ambulances passing through a busy intersection (while on duty) in a large city has a Poisson distribution with mean 0.2 per hour. Assume that the number of ambulances passing through this intersection in any given hour is independent of the number of ambulances in any other hour.

As part of a traffic study, this intersection is observed for two one-hour periods (from 9 AM to 10 AM and 1 PM to 2 PM).

1. What is the probability that there is at least one ambulance passing through the intersection during these two hours?
2. What is the probability that the period of 9 AM to 10 AM has at least one ambulance given that there is at least one ambulance passing through the intersection during these two hours?

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$\copyright \ 2013$