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Chapter 11: Problem 23

The mathematics faculty at a college consists of 8 professors, 12 associate professors, 14 assistant professors, and 10 instructors. If one faculty member is randomly selected, find the probability of choosing a professor or an instructor.

Short Answer

Expert verified
The probability of selecting a professor or an instructor from the faculty is \( \frac{18}{44} \) or approximately 0.409.

Step by step solution

01

Calculation of Total Faculty Members

Add 8 professors, 10 instructors, 12 associate professors, and 14 assistant professors together. This will provide the total number of faculty members.
02

Calculation of Professors and Instructors

Add the number of professors (8) and the number of instructors (10). This total is the number of possible outcomes we are interested in.
03

Calculation of Favorable Probability

Divide the total number of professors and instructors by the total number of faculty members. Present this number as a decimal or a simplified fraction.
04

Final Result

This is the probability of selecting a professor or an instructor from the faculty.

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Key Concepts

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

Faculty Composition
Understanding the makeup of a faculty is essential for solving probability problems like this one. Faculty composition refers to the different categories of teaching staff within an institution. In our example, the mathematics faculty includes:
  • 8 Professors
  • 12 Associate Professors
  • 14 Assistant Professors
  • 10 Instructors
To find the probability of selecting a certain type of faculty member, you first need to know the total number of faculty members. Add all the categories together: 8 + 12 + 14 + 10 equals 44 faculty members. This total is what probability calculations will be based on as it represents all possible outcomes.
Random Selection
Random selection is a key element in probability. It refers to the process of choosing an item from a set where each item has an equal chance of being selected. For our faculty selection process:
  • Each of the 44 faculty members has the same chance of being chosen.
  • No preference or bias is involved in the selection.
This random process ensures fairness and is crucial for calculating the true probability. If one faculty member is chosen randomly, any professor, associate professor, assistant professor, or instructor could be picked, guided only by pure chance.
Event Outcome
In probability, an event refers to the outcome or a combination of outcomes of a random process. Here, the event of interest is picking either a professor or an instructor. Calculation involves identifying:
  • The total outcomes possible: all 44 faculty members.
  • The specific outcomes that make the event happen: 8 professors and 10 instructors.
Adding the 8 professors to the 10 instructors, we see there are 18 favorable outcomes where a professor or an instructor could be chosen. This collection of 18 specific outcomes is what the probability focuses on when determining the likelihood of the event.
Favorable Outcomes
Favorable outcomes are those that meet the criteria set out in the probability question. For this problem:
  • Favorable outcomes include the selection of either a professor or an instructor.
  • There are 18 favorable outcomes (8 professors + 10 instructors).
To find the probability, divide the number of favorable outcomes by the total number of possible outcomes. In this case, it will be \(\frac{18}{44}\), simplified to \(\frac{9}{22}\). This fraction represents the probability of randomly selecting a professor or an instructor from the faculty.

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Most popular questions from this chapter

An oil company is considering two sites on which to drill, described as follows: Site A: Profit if oil is found: \(\$ 80\) million Loss if no oil is found: \(\$ 10\) million Probability of finding oil: \(0.2\) Site B: Profit if oil is found: \(\$ 120\) million Loss if no oil is found: \(\$ 18\) million Probability of finding oil: \(0.1\) Which site has the larger expected profit? By how much?

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