# Exam P Practice Problem 40 – Total Claim Amount

Problem 40-A

The number of claims in a calendar year for an insured has a probability function indicated below.

$\displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Probability} \\\text{ }&\text{ }&\text{ } \\ 0&\text{ }&\displaystyle \frac{27}{64} \\\text{ }&\text{ }&\text{ } \\ 1&\text{ }&\displaystyle \frac{27}{64} \\\text{ }&\text{ }&\text{ } \\ 2&\text{ }&\displaystyle \frac{9}{64} \\\text{ }&\text{ }&\text{ } \\ 3&\text{ }&\displaystyle \frac{1}{64} \end{bmatrix}$

When a claim occurs, the claim amount $X$, regardless of how many claims the insured will have in the calendar year, has probabilities $P(X=1)=0.8$ and $P(X=2)=0.2$. The claim amounts in a calendar year for this insured are independent.

Let $T$ be the total claim amount for this insured in a calendar year. Calculate $P(3 \le T \le 4)$.

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Problem 40-B

A bowl has 3 red balls and 6 white balls. Select two balls at random from this bowl with replacement. Let $N$ be the number of red balls found in the two selected balls. When $N=n$ where $n>0$, roll a fair die $n$ times.

Let $W$ be the sum of the rolls of the die. Calculate $P(4 \le W \le 5)$.

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