# Exam P Practice Problem 67 – Statistical Studies of Insured Drivers

Problem 67-A

An auto insurance company performed a statistical study on its insured drivers. The following table shows the results.

$\displaystyle \begin{bmatrix} \text{Age Group}&\text{ }&\text{Percentage}&\text{ }&\text{Annual Probability of} \\ \text{ }&\text{ }&\text{of its Drivers}&\text{ }&\text{At Least One Claim} \\\text{ }&\text{ }&\text{ } \\ \text{16-20}&\text{ }&15 \% &\text{ }&0.18 \\\text{ }&\text{ }&\text{ } \\ \text{21-30}&\text{ }&20 \% &\text{ }&0.12 \\\text{ }&\text{ }&\text{ } \\ \text{31-50}&\text{ } &30 \% &\text{ }&0.08 \\\text{ }&\text{ }&\text{ } \\ \text{51-70}&\text{ }&25 \% &\text{ }&0.09 \\\text{ }&\text{ }&\text{ } \\ \text{71 and up}&\text{ }&10 \% &\text{ }&0.11 \end{bmatrix}$

The authors of the statistical study also found that for any insured driver in the study, the annual number of claims follows a Poisson distribution.

Suppose that an insured driver in the study had exactly 2 claims in the past year. What is the probability that the insured driver is from the age group 16-20?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.150$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.223$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.249$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.376$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.415$

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Problem 67-B

An auto insurance company performed a statistical study on its younger insured drivers (under 35 years of age). The following table shows the results.

$\displaystyle \begin{bmatrix} \text{Age Group}&\text{ }&\text{Percentage}&\text{ }&\text{Annual Probability of} \\ \text{ }&\text{ }&\text{of its Drivers}&\text{ }&\text{At Least One Claim} \\\text{ }&\text{ }&\text{ } \\ \text{16-17}&\text{ }&12 \% &\text{ }&0.18 \\\text{ }&\text{ }&\text{ } \\ \text{18-24}&\text{ }&38 \% &\text{ }&0.10 \\\text{ }&\text{ }&\text{ } \\ \text{25-34}&\text{ } &50 \% &\text{ }&0.06 \end{bmatrix}$

The authors of the statistical study also found that for any insured driver in the study, the annual number of claims follows a Poisson distribution. Furthermore, for any insured driver in the study, the number of claims in one year is independent of the number of claims in any other year.

Suppose that in a 2-year period, an insured driver in the study had exactly 1 claim in year 1 and exactly 2 claims in year 2. What is the probability that the insured driver is from the age group 16-17?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.120$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.229$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.241$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.329$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.576$

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