# Exam P Practice Problem 62 – Waiting for Telephone Calls

Problem 62-A

An individual classifies the telephone calls he receives into the categories of Personal Calls (e.g. calls from friends and relatives) and Non-Personal Calls (all the other calls that are considered not Personal Calls).

Let $X$ be the time (in minutes) until the next Personal Call. Let $Y$ be the time (in minutes) until the next Non-Personal Call.

Suppose that $X$ and $Y$ are independent random variables and follow exponential distributions with means 8 minutes and 3 minutes, respectively.

What is the probability that the next incoming telephone call is a Personal Call?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2727$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3735$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.5000$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6265$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.7273$

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Problem 62-B

An individual classifies the telephone calls he receives into the categories of Personal Calls (e.g. calls from friends and relatives), Business Calls (calls related to his small business) and Other Calls (all the other calls not belonging to the Personal Call and Business Call categories).

Let $X$ be the time (in minutes) until the next Personal Call, let $Y$ be the time (in minutes) until the next Business Call and let $Z$ be the time (in minutes) until the next Other Call.

Suppose $X$, $Y$ and $Z$ are independent random variables and follow exponential distributions with means 12, 10 and 6 minutes, respectively.

What is the probability that the next telephone call this individual receives will be a Business Call?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{3}{14}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{4}{14}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{3}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{14}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{6}{14}$

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Answers

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