Exam P Practice Problem 110 – likelihood of auto accidents
Problem 110A
An actuary studied the likelihood of accidents in a oneyear 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 

1620  15%  0.20  0.25 
2130  25%  0.35  0.40 
3150  35%  0.60  0.30 
5170  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 2130.
Problem 110B
An auto insurance company performed a study on the frequency of accidents of its insured drivers in a oneyear period. The following table gives the results of the study.
Age Group  Percent of Drivers  Probability of At Least 1 Accident 

1620  10%  0.30 
2140  20%  0.20 
4165  35%  0.10 
66+  35%  0.12 
A randomly selected insured driver from the study was found to have no accidents in the oneyear period.
Calculate the probability that the insured driver is from the age group 1620.
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Exam P Practice Problem 109 – counting insurance payments
Problem 109A
Amounts of damages due to auto collision accidents follow a probability distribution whose density function is given by the following.
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.
Problem 109B
Amounts of damages due to auto collision accidents follow a probability distribution whose density function is given by the following.
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.
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Exam P Practice Problem 102 – estimating claim costs
Problem 102A
Insurance claims modeled by a distribution with the following cumulative distribution function.
The insurance company is performing a study on all claims that exceed 3. Determine the mean of all claims being studied.
Problem 102B
Insurance claims modeled by a distribution with the following cumulative distribution function.
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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Exam P Practice Problem 99 – When Random Loss is Doubled
Problem 99A
A business owner faces a risk whose economic loss amount follows a uniform distribution over the interval . In the next year, the loss amount is expected to be doubled and is expected to be modeled by the random variable .
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.
Problem 99B
A business owner faces a risk whose economic loss amount has the following density function:
In the next year, the loss amount is expected to be doubled and is expected to be modeled by the random variable .
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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Exam P Practice Problem 97 – Variance of Claim Sizes
Problem 97A
For a type of insurance policies, the following is the probability that the size of claim is greater than .
Calculate the variance of the claim size for this type of insurance policies.
Problem 97B
For a type of insurance policies, the following is the probability that the size of a claim is greater than .
Calculate the expected claim size for this type of insurance policies.
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Exam P Practice Problem 96 – Expected Insurance Payment
Problem 96A
An insurance policy is purchased to cover a random loss subject to a deductible of 1. The cumulative distribution function of the loss amount is:
Given a random loss , determine the expected payment made under this insurance policy.
Problem 96B
An insurance policy is purchased to cover a random loss subject to a deductible of 2. The density function of the loss amount is:
Given a random loss , what is the expected benefit paid by this insurance policy?
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Exam P Practice Problem 93 – Determining Average Claim Frequency
Problem 93A
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 where . Furthermore, she finds that is proportional to the following function:
What is the weekly average number of claims arising from this group of insurance policies?
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Problem 93B
Let 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 is proportional to the following function:
What is the average number of taxis arriving at this airport terminal per minute?
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Exam P Practice Problem 92 – Expected Claim Payment
Problem 92A
The size of a claim that an auto insurance company pays out is modeled by a random variable with the following density function:
By subjecting the insured to a deductible of 12 per claim, what is the expected reduction in claim payment?
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Problem 92B
The size of a claim that an auto insurance company pays out is modeled by a random variable with the following density function:
By subjecting the insured to a deductible of 10 per claim, by what percent is the expected claim payment reduced?
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Exam P Practice Problem 90 – Insurance Benefits
Problem 90A
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?
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Problem 90B
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?
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