91Ó°ÊÓ

A technician services mailing machines at companies in the Phoenix area. Depending on the type of malfunction, the service call can take \(1,2,3,\) or 4 hours. The different types of malfunctions occur at about the same frequency. a. Develop a probability distribution for the duration of a service call. b. Draw a graph of the probability distribution. c. Show that your probability distribution satisfies the conditions required for a discrete probability function. d. What is the probability a service call will take three hours? e. A service call has just come in, but the type of malfunction is unknown. It is 3: 00 P.M. and service technicians usually get off at 5: 00 P.M. What is the probability the service technician will have to work overtime to fix the machine today?

Step by step solution

01

Understand the Situation

The technician deals with mailing machines having malfunctions that take different durations to fix: 1, 2, 3, or 4 hours. Each type of malfunction occurs with equal frequency.

02

Develop Probability Distribution

To create a probability distribution, assign a probability to each service duration. Since there are four equally likely durations (1, 2, 3, or 4 hours), assign each a probability of \( \frac{1}{4} \). Thus, \( P(X=1) = \frac{1}{4} \), \( P(X=2) = \frac{1}{4} \), \( P(X=3) = \frac{1}{4} \), and \( P(X=4) = \frac{1}{4} \).

03

Graph the Probability Distribution

Create a bar graph with the x-axis representing the different durations (1 to 4 hours) and the y-axis representing the probability (\( \frac{1}{4} \)). Each duration has a bar reaching up to \( \frac{1}{4} \) on the y-axis.

04

Validate Probability Distribution

A discrete probability distribution must have probabilities that sum up to 1. Calculate the sum: \( \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1 \). Also, each probability is between 0 and 1. Both conditions are satisfied.

05

Probability of Three-Hour Call

Find the probability that a service call takes exactly 3 hours. From the distribution: \( P(X=3) = \frac{1}{4} \).

06

Overtime Probability Calculation

Calculate the probability that the service time exceeds 2 hours (since they start at 3:00 PM and usually leave at 5:00 PM). The overtime event includes \( X = 3 \) hours or \( X = 4 \) hours: \( P(X > 2) = P(X=3) + P(X=4) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Probability Function

In the realm of probability and statistics, a **discrete probability function** is a fundamental concept that assigns probabilities to discrete outcomes. Discrete outcomes are those that can be counted, like the duration of service calls, each lasting a specific number of hours. To construct a discrete probability function for the scenario described:

• Identify all possible outcomes: in this case, the service call durations of 1, 2, 3, and 4 hours.
• Since each duration is equally likely, assign each a probability of \( \frac{1}{4} \).
• Ensure the sum of all probabilities is 1, fulfilling the condition that total probability must be 100%.

Here, the probability function would be: \( P(X=1) = \frac{1}{4} \), \( P(X=2) = \frac{1}{4} \), \( P(X=3) = \frac{1}{4} \), \( P(X=4) = \frac{1}{4} \). Each of these probabilities is valid because it lies between 0 and 1, and their sum is 1.

Probability Graph

Visual representation of probabilities can significantly aid understanding. This is where a **probability graph**, such as a bar chart, comes into play. It provides a straightforward way to illustrate the probability distribution of a discrete random variable.For our mailing machine scenario:

• The x-axis represents the different possible durations: 1 hour, 2 hours, 3 hours, and 4 hours.
• The y-axis reflects the probability of each duration, which in this case is consistently \( \frac{1}{4} \).
• Each duration corresponds to a bar reaching a height of \( \frac{1}{4} \).

Such a bar graph clearly shows the uniform distribution of probabilities across possible service durations, aiding in both calculation and conceptual understanding.

Discrete Random Variable

A **discrete random variable** is a variable that can take on only a countable number of values. It forms the basis for discrete probability functions and graphs. Unlike continuous variables, which can assume any value in an interval, discrete random variables are limited to specific values. In the example, the duration of a service call is a discrete random variable with possible values of 1, 2, 3, or 4 hours. Here's how it works:

• Each value corresponds to a specific event (the completion of a service call).
• The probabilities are predefined and fixed based on the occurrence frequency.
• Because it's discrete, you can count the potential outcomes, making calculations straightforward.

Understanding discrete random variables is essential as they form the underpinnings of basic probability calculations, like measuring the chances of a service call of certain duration, or predicting service needs in different time frames.