Discover a wealth of knowledge and get your questions answered on IDNLearn.com. Ask your questions and receive detailed and reliable answers from our experienced and knowledgeable community members.
Sagot :
To address this question comprehensively, let's detail the process step by step:
1. Understand the Original Situation:
We have a school where students are taking either Biology, Textiles, both, or neither of the subjects. Initially, the probability of a student taking Biology is [tex]\(\frac{7}{20}\)[/tex].
2. Determine the Initial Number of Students:
Given:
- Probability of taking Biology ([tex]\(P(B)\)[/tex]) = [tex]\(\frac{7}{20}\)[/tex]
- Let's denote the initial number of students as [tex]\(n_{initial}\)[/tex].
3. Set Up the Equation with Initial Probability:
- The number of students taking Biology initially is [tex]\((\frac{7}{20}) \times n_{initial}\)[/tex].
- When a new student joins, the probability of taking Biology changes to [tex]\(\frac{1}{3}\)[/tex].
4. Set Up the Equation with New Probability:
- The new total number of students is [tex]\(n_{initial} + 1\)[/tex].
- The new number of students taking Biology remains the same count as initially but spread over the new total number. So, it will be [tex]\(\frac{1}{3} \times (n_{initial} + 1)\)[/tex].
5. Solve the Equation:
- Equate the two expressions for the number of students taking Biology:
[tex]\[ (\frac{7}{20}) \times n_{initial} = \frac{1}{3} \times (n_{initial} + 1) \][/tex]
- Solving this equation gives us:
[tex]\(n_{initial} = 20\)[/tex].
6. Calculate Initial Number of Students Taking Biology:
- Number of students initially taking Biology: [tex]\( (\frac{7}{20}) \times 20 = 7 \)[/tex].
7. Calculate Students Taking Textiles Initially:
- In the question, it is given that 3 students take neither subject.
- Therefore, the students taking Textiles initially will be:
[tex]\[ n_{initial} - (students_{biology} + students_{neither}) = 20 - (7 + 3) = 10 \][/tex]
8. Calculate the Number of Students After the New Student Joins:
- The total number of students after the new student joins: [tex]\(n_{initial} + 1 = 20 + 1 = 21\)[/tex].
- The number of students taking Biology remains 7.
- The number of students taking Textiles remains the same as well: 10.
- The number of students taking neither remains: 3.
To represent this information on a Venn diagram:
- The circle for Biology will have 7 students.
- The circle for Textiles will have 10 students.
- The area outside both circles will have 3 students.
This results in the populations being broken down and verified appropriately for both the initial and new scenarios.
1. Understand the Original Situation:
We have a school where students are taking either Biology, Textiles, both, or neither of the subjects. Initially, the probability of a student taking Biology is [tex]\(\frac{7}{20}\)[/tex].
2. Determine the Initial Number of Students:
Given:
- Probability of taking Biology ([tex]\(P(B)\)[/tex]) = [tex]\(\frac{7}{20}\)[/tex]
- Let's denote the initial number of students as [tex]\(n_{initial}\)[/tex].
3. Set Up the Equation with Initial Probability:
- The number of students taking Biology initially is [tex]\((\frac{7}{20}) \times n_{initial}\)[/tex].
- When a new student joins, the probability of taking Biology changes to [tex]\(\frac{1}{3}\)[/tex].
4. Set Up the Equation with New Probability:
- The new total number of students is [tex]\(n_{initial} + 1\)[/tex].
- The new number of students taking Biology remains the same count as initially but spread over the new total number. So, it will be [tex]\(\frac{1}{3} \times (n_{initial} + 1)\)[/tex].
5. Solve the Equation:
- Equate the two expressions for the number of students taking Biology:
[tex]\[ (\frac{7}{20}) \times n_{initial} = \frac{1}{3} \times (n_{initial} + 1) \][/tex]
- Solving this equation gives us:
[tex]\(n_{initial} = 20\)[/tex].
6. Calculate Initial Number of Students Taking Biology:
- Number of students initially taking Biology: [tex]\( (\frac{7}{20}) \times 20 = 7 \)[/tex].
7. Calculate Students Taking Textiles Initially:
- In the question, it is given that 3 students take neither subject.
- Therefore, the students taking Textiles initially will be:
[tex]\[ n_{initial} - (students_{biology} + students_{neither}) = 20 - (7 + 3) = 10 \][/tex]
8. Calculate the Number of Students After the New Student Joins:
- The total number of students after the new student joins: [tex]\(n_{initial} + 1 = 20 + 1 = 21\)[/tex].
- The number of students taking Biology remains 7.
- The number of students taking Textiles remains the same as well: 10.
- The number of students taking neither remains: 3.
To represent this information on a Venn diagram:
- The circle for Biology will have 7 students.
- The circle for Textiles will have 10 students.
- The area outside both circles will have 3 students.
This results in the populations being broken down and verified appropriately for both the initial and new scenarios.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.
