# Exam P Practice Problem 70 – Real Estate Sales Contest

Problem 70-A

A commercial real estate property company has three sales agents who are actively selling commercial real estate properties. The times (in days) to the next successful sale for these three agents are exponentially distributed with means 10 days, 15 days and 20 days.

These three agents work independently. So the time to the next successful sale for one agent is independent of the time to the next successful sale for any of the other agents.

To spur sales, the company has a contest among the three agents. Each agent produces a sale. The award will go to the first agent producing the first sale.

What is the probability that the winning sale will take place within one week?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.14$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.22$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.50$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.78$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.86$

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Problem 70-B

A commercial real estate property company has four sales agents who are actively selling commercial real estate properties. The times (in days) to the next successful sale for these four agents are exponentially distributed with means 10 days, 15 days and 20 days and 30 days.

These four agents work independently. So the time to the next successful sale for one agent is independent of the time to the next successful sale for any of the other agents.

To spur sales, the company has a contest among the four agents. Each agent produces a sale. The award will go to the first agent producing the first sale.

What is the expected waiting time (in days) from the beginning of the contest to the occurrence of the winning sale?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10$

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