Answer :
To solve the problem "Assuming a student is taking a foreign language, what is the probability the student is also in high school?", let's break down the information step-by-step using the provided data.
Step 1: Understand the given table
We have data on students taking and not taking a foreign language in both middle school and high school.
Here’s the table for reference:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Taking a Foreign Language} & \text{Not Taking a Foreign Language} & \text{Total} \\ \hline \text{Middle School} & 0.68 & 0.32 & 1.0 \\ \hline \text{High School} & 0.88 & 0.12 & 1.0 \\ \hline \text{Total} & 0.8 & 0.2 & 1.0 \\ \hline \end{array} \][/tex]
Step 2: Extract necessary probabilities
- The probability that a student is in high school and is taking a foreign language is 0.88 (from the High School row, Taking a Foreign Language column).
- The total probability that a student is taking a foreign language is 0.8 (from the Total row, Taking a Foreign Language column).
Step 3: Calculate the joint probability
To find the probability that a student is both in high school and taking a foreign language:
[tex]\[ P(\text{Taking a Foreign Language} \cap \text{High School}) = 0.88 \times 0.2 = 0.176 \][/tex]
Step 4: Calculate the desired conditional probability
We need to find [tex]\( P(\text{High School} \mid \text{Taking a Foreign Language}) \)[/tex]. The conditional probability formula is:
[tex]\[ P(\text{High School} \mid \text{Taking a Foreign Language}) = \frac{P(\text{Taking a Foreign Language} \cap \text{High School})}{P(\text{Taking a Foreign Language})} \][/tex]
Using the numbers we have:
[tex]\[ P(\text{High School} \mid \text{Taking a Foreign Language}) = \frac{0.176}{0.8} = 0.22 \][/tex]
Conclusion:
- Probability that a student is taking a foreign language and is in high school: 0.176
- Probability that a student is in high school given they are taking a foreign language: 0.22
Answering the specific question:
"Table [tex]\( B \)[/tex], because the given condition is that the student is taking a foreign language."
This is correct because Table [tex]\( B \)[/tex] presents the frequencies needed to determine conditional probabilities involving students taking foreign languages and being in high school or middle school.
Step 1: Understand the given table
We have data on students taking and not taking a foreign language in both middle school and high school.
Here’s the table for reference:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Taking a Foreign Language} & \text{Not Taking a Foreign Language} & \text{Total} \\ \hline \text{Middle School} & 0.68 & 0.32 & 1.0 \\ \hline \text{High School} & 0.88 & 0.12 & 1.0 \\ \hline \text{Total} & 0.8 & 0.2 & 1.0 \\ \hline \end{array} \][/tex]
Step 2: Extract necessary probabilities
- The probability that a student is in high school and is taking a foreign language is 0.88 (from the High School row, Taking a Foreign Language column).
- The total probability that a student is taking a foreign language is 0.8 (from the Total row, Taking a Foreign Language column).
Step 3: Calculate the joint probability
To find the probability that a student is both in high school and taking a foreign language:
[tex]\[ P(\text{Taking a Foreign Language} \cap \text{High School}) = 0.88 \times 0.2 = 0.176 \][/tex]
Step 4: Calculate the desired conditional probability
We need to find [tex]\( P(\text{High School} \mid \text{Taking a Foreign Language}) \)[/tex]. The conditional probability formula is:
[tex]\[ P(\text{High School} \mid \text{Taking a Foreign Language}) = \frac{P(\text{Taking a Foreign Language} \cap \text{High School})}{P(\text{Taking a Foreign Language})} \][/tex]
Using the numbers we have:
[tex]\[ P(\text{High School} \mid \text{Taking a Foreign Language}) = \frac{0.176}{0.8} = 0.22 \][/tex]
Conclusion:
- Probability that a student is taking a foreign language and is in high school: 0.176
- Probability that a student is in high school given they are taking a foreign language: 0.22
Answering the specific question:
"Table [tex]\( B \)[/tex], because the given condition is that the student is taking a foreign language."
This is correct because Table [tex]\( B \)[/tex] presents the frequencies needed to determine conditional probabilities involving students taking foreign languages and being in high school or middle school.
