Among 500 freshmen pursuing a business degree at a university, 338 are enrolled in an economics course, 220 are enrolled in a mathematics course, and 123 are enrolled in both an economics and a mathematics course. What is the probability that a freshman selected at random from this group is enrolled in each of the following? (Enter your answers to three decimal places.) (a) an economics and/or a mathematics course (b) exactly one of these two courses (c) neither an economics course nor a mathematics course
Among 500 freshmen pursuing a business degree at a university, 338 are enrolled in an economics course, 220 are enrolled in a mathematics course, and 123 are enrolled in both an economics and a mathematics course. What is the probability that a freshman selected at random from this group is enrolled in each of the following? (Enter your answers to three decimal places.) (a) an economics and/or a mathematics course (b) exactly one of these two courses (c) neither an economics course nor a mathematics course
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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![**Probability and Set Theory in University Course Enrollment**
Let's explore a practical application of probability and set theory among university students.
**Problem Statement:**
Among 500 freshmen pursuing a business degree at a university, 338 are enrolled in an economics course, 220 are enrolled in a mathematics course, and 123 are enrolled in both an economics and a mathematics course. What is the probability that a freshman selected at random from this group is enrolled in each of the following? (Enter your answers to three decimal places):
(a) An economics and/or a mathematics course
(b) Exactly one of these two courses
(c) Neither an economics course nor a mathematics course
**Solution Approach:**
To solve these problems, we will use the principle of inclusion and exclusion for sets and basic probability formulas.
**(a) Probability of Enrolled in an Economics and/or a Mathematics Course:**
The probability of being enrolled in an economics and/or a mathematics course is given by:
\[ P(E \cup M) = P(E) + P(M) - P(E \cap M) \]
Where:
- \( P(E) \) is the probability of being enrolled in an economics course
- \( P(M) \) is the probability of being enrolled in a mathematics course
- \( P(E \cap M) \) is the probability of being enrolled in both courses
Given:
- \( P(E) = \frac{338}{500} \)
- \( P(M) = \frac{220}{500} \)
- \( P(E \cap M) = \frac{123}{500} \)
Thus,
\[ P(E \cup M) = \frac{338}{500} + \frac{220}{500} - \frac{123}{500} \]
\[ P(E \cup M) = \frac{338 + 220 - 123}{500} \]
\[ P(E \cup M) = \frac{435}{500} \]
\[ P(E \cup M) = 0.870\]
**(b) Probability of Enrolled in Exactly One of the Two Courses:**
The probability of being enrolled in exactly one of the two courses is given by:
\[ P(\text{Exactly one}) = P(E \setminus M) + P(M \setminus E) \]
Where:
- \( P(E \setminus M) \) is the probability of being enrolled in only an economics course
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Transcribed Image Text:**Probability and Set Theory in University Course Enrollment**
Let's explore a practical application of probability and set theory among university students.
**Problem Statement:**
Among 500 freshmen pursuing a business degree at a university, 338 are enrolled in an economics course, 220 are enrolled in a mathematics course, and 123 are enrolled in both an economics and a mathematics course. What is the probability that a freshman selected at random from this group is enrolled in each of the following? (Enter your answers to three decimal places):
(a) An economics and/or a mathematics course
(b) Exactly one of these two courses
(c) Neither an economics course nor a mathematics course
**Solution Approach:**
To solve these problems, we will use the principle of inclusion and exclusion for sets and basic probability formulas.
**(a) Probability of Enrolled in an Economics and/or a Mathematics Course:**
The probability of being enrolled in an economics and/or a mathematics course is given by:
\[ P(E \cup M) = P(E) + P(M) - P(E \cap M) \]
Where:
- \( P(E) \) is the probability of being enrolled in an economics course
- \( P(M) \) is the probability of being enrolled in a mathematics course
- \( P(E \cap M) \) is the probability of being enrolled in both courses
Given:
- \( P(E) = \frac{338}{500} \)
- \( P(M) = \frac{220}{500} \)
- \( P(E \cap M) = \frac{123}{500} \)
Thus,
\[ P(E \cup M) = \frac{338}{500} + \frac{220}{500} - \frac{123}{500} \]
\[ P(E \cup M) = \frac{338 + 220 - 123}{500} \]
\[ P(E \cup M) = \frac{435}{500} \]
\[ P(E \cup M) = 0.870\]
**(b) Probability of Enrolled in Exactly One of the Two Courses:**
The probability of being enrolled in exactly one of the two courses is given by:
\[ P(\text{Exactly one}) = P(E \setminus M) + P(M \setminus E) \]
Where:
- \( P(E \setminus M) \) is the probability of being enrolled in only an economics course
-
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