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A discrete probability function is a fundamental concept in probability theory. It gives us a way to assign probabilities to outcomes in a scenario where outcomes are countable and distinct. In this exercise, the duration of a service call is given as either 1, 2, 3, or 4 hours. This means the outcomes are finite, making it a perfect example of a discrete setting.

To develop a discrete probability function, each distinct outcome must be assigned a probability. Here, all outcomes (1, 2, 3, and 4 hours) are equally likely because each malfunction type occurs with the same frequency. Thus, the probability for each is calculated by dividing 1 by the number of outcomes. Since there are 4 possible outcomes, the probability of each is \( \frac{1}{4} = 0.25 \). This ensures that the total probability across all possible outcomes is 1, satisfying one of the key conditions of a valid discrete probability function.

Remember, when constructing a probability distribution function for discrete variables, it's essential that:

• All probabilities must lie between 0 and 1.
• The sum of all probabilities equals 1.

A probability graph provides a visual representation of a probability distribution. For discrete distributions, one of the simplest ways to create this graph is by using a bar graph. In our current exercise, the probability graph illustrates the likelihood of different durations for a service call.

To construct a bar graph in this context:

• The x-axis represents the duration of the service call (1, 2, 3, or 4 hours).
• The y-axis shows the probability associated with each duration, which is 0.25 in this case.
• Beneath each duration value on the x-axis, place a vertical bar with a height corresponding to its probability (0.25).

This approach visually reinforces the equal probability of all durations. The heights of the bars make it immediately clear that each duration is equally likely. Good graphs are not just about correctness but also about clear and effective communication of the data they present.

A probability table is an organized way to present data, displaying variables and their corresponding probabilities. In our exercise, the probability table captures the information about how long a service call might take and the probability of each duration.

This table is set up as follows:

• The first column lists the possible outcomes, which in this case, are service call durations: 1, 2, 3, and 4 hours.
• The second column lists the probabilities associated with each outcome: all are 0.25.

Such a table not only simplifies the information but also highlights the key characteristics of a discrete probability distribution. Namely, it is easy to see that each event has an equal probability and that together, the probabilities sum to 1. This reinforces the comprehensive nature of the probability function.

Understanding probability tables is crucial because they provide a quick reference for decision-making, such as determining the likelihood of various outcomes. Additionally, they serve as a foundational tool for building more advanced statistical models.