Suppose a math class contains 42 students, 18 females (six of whom speak French) and 24 males (four of whom speak French). Compute the probability that a randomly selected student is female, given that the student speaks French.
Suppose a math class contains 42 students, 18 females (six of whom speak French) and 24 males (four of whom speak French). Compute the probability that a randomly selected student is female, given that the student speaks French.
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Counting And Probability
Section: Chapter Questions
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![**Conditional Probability in a Math Class**
In this example, we will explore conditional probability in the context of a math class.
**Problem Statement:**
Suppose a math class contains 42 students, with details as follows:
- There are 18 females.
- Out of these 18 females, 6 speak French.
- There are 24 males.
- Out of these 24 males, 4 speak French.
The total number of students who speak French (both male and female) is 10.
We need to compute the probability that a randomly selected student is female, given that the student speaks French.
**Let's break down the problem:**
1. **Total number of students:** 42
2. **Total number of females:** 18
3. **Total number of males:** 24
4. **Number of females who speak French:** 6
5. **Number of males who speak French:** 4
6. **Total number of students who speak French:** 6 (females) + 4 (males) = 10
**Approach to solve:**
To find the probability that a randomly selected student is female, given that the student speaks French, we use the formula for conditional probability:
\[ P(\text{Female} | \text{Speak French}) = \frac{P(\text{Female and Speak French})}{P(\text{Speak French})} \]
where:
- \( P(\text{Female and Speak French}) \) is the probability that a student is both female and speaks French.
- \( P(\text{Speak French}) \) is the probability that a student speaks French.
**Calculations:**
1. \( P(\text{Female and Speak French}) = \frac{\text{Number of females who speak French}}{\text{Total number of students}} = \frac{6}{42} = \frac{1}{7} \)
2. \( P(\text{Speak French}) = \frac{\text{Total number of students who speak French}}{\text{Total number of students}} = \frac{10}{42} = \frac{5}{21} \)
So,
\[ P(\text{Female} | \text{Speak French}) = \frac{P(\text{Female and Speak French})}{P(\text{Speak French})} = \frac{\frac{1}{7}}{\frac{5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52bde6f1-de69-4d5b-bf13-f82b423cf0e9%2Fa9f442c2-8387-4aee-98bb-f9bf87a8721d%2F7y1w9n_processed.png&w=3840&q=75)
Transcribed Image Text:**Conditional Probability in a Math Class**
In this example, we will explore conditional probability in the context of a math class.
**Problem Statement:**
Suppose a math class contains 42 students, with details as follows:
- There are 18 females.
- Out of these 18 females, 6 speak French.
- There are 24 males.
- Out of these 24 males, 4 speak French.
The total number of students who speak French (both male and female) is 10.
We need to compute the probability that a randomly selected student is female, given that the student speaks French.
**Let's break down the problem:**
1. **Total number of students:** 42
2. **Total number of females:** 18
3. **Total number of males:** 24
4. **Number of females who speak French:** 6
5. **Number of males who speak French:** 4
6. **Total number of students who speak French:** 6 (females) + 4 (males) = 10
**Approach to solve:**
To find the probability that a randomly selected student is female, given that the student speaks French, we use the formula for conditional probability:
\[ P(\text{Female} | \text{Speak French}) = \frac{P(\text{Female and Speak French})}{P(\text{Speak French})} \]
where:
- \( P(\text{Female and Speak French}) \) is the probability that a student is both female and speaks French.
- \( P(\text{Speak French}) \) is the probability that a student speaks French.
**Calculations:**
1. \( P(\text{Female and Speak French}) = \frac{\text{Number of females who speak French}}{\text{Total number of students}} = \frac{6}{42} = \frac{1}{7} \)
2. \( P(\text{Speak French}) = \frac{\text{Total number of students who speak French}}{\text{Total number of students}} = \frac{10}{42} = \frac{5}{21} \)
So,
\[ P(\text{Female} | \text{Speak French}) = \frac{P(\text{Female and Speak French})}{P(\text{Speak French})} = \frac{\frac{1}{7}}{\frac{5
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