# 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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