Suppose a math class contains 41 students, 19 females (two of whom speak French) and 22 males (three of whom speak French). Compute the probability that a randomly selected student is male, given that the student speaks French.
Suppose a math class contains 41 students, 19 females (two of whom speak French) and 22 males (three of whom speak French). Compute the probability that a randomly selected student is male, given that the student speaks French.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
![**Problem Statement: Conditional Probability**
Suppose a math class contains 41 students, 19 females (two of whom speak French) and 22 males (three of whom speak French). Compute the probability that a randomly selected student is male, given that the student speaks French.
**Solution:**
Let:
- F denote the event that a student is female.
- M denote the event that a student is male.
- S denote the event that a student speaks French.
Given:
- Total number of students = 41
- Number of female students = 19
- Number of male students = 22
- Number of French-speaking female students = 2
- Number of French-speaking male students = 3
We need to determine \( P(M|S) \), the probability that a student is male given that the student speaks French.
First, we calculate the total number of French-speaking students:
\[
\text{Number of French-speaking students} = 2 + 3 = 5
\]
Using the definition of conditional probability:
\[
P(M|S) = \frac{P(M \cap S)}{P(S)}
\]
Where:
- \( P(M \cap S) \) is the probability that a student is male and speaks French.
- \( P(S) \) is the probability that a student speaks French.
Calculate \( P(M \cap S) \):
\[
P(M \cap S) = \frac{\text{Number of French-speaking male students}}{\text{Total number of students}} = \frac{3}{41}
\]
Calculate \( P(S) \):
\[
P(S) = \frac{\text{Total number of French-speaking students}}{\text{Total number of students}} = \frac{5}{41}
\]
Substitute these values into the conditional probability formula:
\[
P(M|S) = \frac{\frac{3}{41}}{\frac{5}{41}} = \frac{3}{5}
\]
Therefore, the probability that a randomly selected student is male, given that the student speaks French, is \( \frac{3}{5} \) or 0.6.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f8e4cad-17a8-4afc-8111-d96bcef8136e%2Ffe8786ed-1ba0-475e-9cac-9d6a2a3500e3%2Fotnpr6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Conditional Probability**
Suppose a math class contains 41 students, 19 females (two of whom speak French) and 22 males (three of whom speak French). Compute the probability that a randomly selected student is male, given that the student speaks French.
**Solution:**
Let:
- F denote the event that a student is female.
- M denote the event that a student is male.
- S denote the event that a student speaks French.
Given:
- Total number of students = 41
- Number of female students = 19
- Number of male students = 22
- Number of French-speaking female students = 2
- Number of French-speaking male students = 3
We need to determine \( P(M|S) \), the probability that a student is male given that the student speaks French.
First, we calculate the total number of French-speaking students:
\[
\text{Number of French-speaking students} = 2 + 3 = 5
\]
Using the definition of conditional probability:
\[
P(M|S) = \frac{P(M \cap S)}{P(S)}
\]
Where:
- \( P(M \cap S) \) is the probability that a student is male and speaks French.
- \( P(S) \) is the probability that a student speaks French.
Calculate \( P(M \cap S) \):
\[
P(M \cap S) = \frac{\text{Number of French-speaking male students}}{\text{Total number of students}} = \frac{3}{41}
\]
Calculate \( P(S) \):
\[
P(S) = \frac{\text{Total number of French-speaking students}}{\text{Total number of students}} = \frac{5}{41}
\]
Substitute these values into the conditional probability formula:
\[
P(M|S) = \frac{\frac{3}{41}}{\frac{5}{41}} = \frac{3}{5}
\]
Therefore, the probability that a randomly selected student is male, given that the student speaks French, is \( \frac{3}{5} \) or 0.6.
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