Category Archives: Negative Binomial Distribution

Exam P Practice Problem 60 – Health Insurance Claim Frequency

Problem 60-A

An insurance company issued health insurance policies to individuals. The company determined that Y, the number of claims filed by an insured in a year, is a random variable with the following probability function.

      \displaystyle P(Y=y)=0.45 \ (0.55)^{\displaystyle y} \ \ \ \ \ \ y=0,1,2,3,\cdots

What is the probability that a random selected insured from this group of insured individuals will file more than 5 claims in a year?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0226

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0277

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0357

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0503

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0749

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Problem 60-B

An insurance company issued health insurance policies to individuals. The company determined that Y, the number of claims filed by an insured in a year, is a random variable with the following probability function.

      \displaystyle P(Y=y)=0.45 \ (0.55)^{\displaystyle y} \ \ \ \ \ \ y=0,1,2,3,\cdots

The number of claims filed by one insured individual is independent of the number of claims filed by any other insured individual.

An actuary studied three randomly selected insured individuals from this group of individuals who purchased health policies from this company. What is the probability that these three insured individuals will file more than 6 claims in a year?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0457

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0706

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1495

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2201

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2406

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Exam P Practice Problem 24 – Total Number of Claims

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Problem 24A
An insurance portfolio consists of four policyholders. The number of claims N for each policyholder in a calendar year follows a distribution with the following probability function:

    \displaystyle P(N=k)=\biggl( \frac{13}{25} \biggr)^k \ \frac{12}{25} \ \ \ \ \ \ \ \ k=0,1,2,3,\cdots

Assume that the number of claims for one policyholder is independent of the number of claims for any one of the other policyholders in the portfolio. What is the probability that the total number of claims in this portfolio in the upcoming calendar year is 4?

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Problem 24B
Use the same information as in Problem 24A. What is the probability that there will be at most 4 claims in the portfolio in the upcoming year?

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Exam P Practice Problem 23 – Medical Screening

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Problem 23A
A researcher in a medical study screens prospective participants for having high serum cholesterol. Let Y represent the number of prospective participants screened until the fifth person wtih high serum cholesterol is found. It is known from past experience that the expected number of prospective participants screened until finding the fifth person with high serum cholesterol is 25. The researcher wants to know the likelihood of achieving his objective by screening just 12 to 15 prospective participants.

What is the probability of finding 5 participants with high serum cholesterol by screening 12 to 15 prospective participants (inclusive for both endpoints)?

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Problem 23B
One ball is drawn successively with replacement from an urn containing one red ball and three white balls. Find the probability that at most 6 draws are necessary to obtain 3 red balls.

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