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.
