Answer :
To solve this problem step-by-step, let's break it down into its respective parts.
Step 1: Define the Variables
Given the marks and the number of students:
- Marks distribution:
- Students who scored 1 mark: [tex]\( n_1 = m + 2 \)[/tex]
- Students who scored 2 marks: [tex]\( n_2 = m - 1 \)[/tex]
- Students who scored 3 marks: [tex]\( n_3 = 2m - 3 \)[/tex]
- Students who scored 4 marks: [tex]\( n_4 = m + 5 \)[/tex]
- Students who scored 5 marks: [tex]\( n_5 = 3m - 4 \)[/tex]
Step 2: Calculate the Total Number of Students
Sum of all students:
[tex]\[ n_1 + n_2 + n_3 + n_4 + n_5 \][/tex]
[tex]\[ (m + 2) + (m - 1) + (2m - 3) + (m + 5) + (3m - 4) \][/tex]
[tex]\[ = m + 2 + m - 1 + 2m - 3 + m + 5 + 3m - 4 \][/tex]
[tex]\[ = 8m - 1 \][/tex]
Step 3: Calculate the Total Marks
Total marks scored by students:
[tex]\[ 1 \cdot n_1 + 2 \cdot n_2 + 3 \cdot n_3 + 4 \cdot n_4 + 5 \cdot n_5 \][/tex]
[tex]\[ = 1(m + 2) + 2(m - 1) + 3(2m - 3) + 4(m + 5) + 5(3m - 4) \][/tex]
[tex]\[ = m + 2 + 2m - 2 + 6m - 9 + 4m + 20 + 15m - 20 \][/tex]
[tex]\[ = 28m - 9 \][/tex]
Step 4: Set Up the Mean Equation
The mean mark is given as [tex]\( \frac{36}{23} \)[/tex].
Equation for the mean:
[tex]\[ \frac{\text{Total Marks}}{\text{Total Students}} = \frac{36}{23} \][/tex]
[tex]\[ \frac{28m - 9}{8m - 1} = \frac{36}{23} \][/tex]
Step 5: Solve for m
Cross-multiplying to find the value of [tex]\( m \)[/tex]:
[tex]\[ 23(28m - 9) = 36(8m - 1) \][/tex]
[tex]\[ 644m - 207 = 288m - 36 \][/tex]
[tex]\[ 644m - 288m = -36 + 207 \][/tex]
[tex]\[ 356m = 171 \][/tex]
[tex]\[ m = \frac{171}{356} \][/tex]
Calculating, we get:
[tex]\[ m = 0.480337078651685 \][/tex]
Step 6: Calculate Total Number of Students
Substitute [tex]\( m \)[/tex] back into the total students' equation:
[tex]\[ 8m - 1 \][/tex]
[tex]\[ 8(0.480337078651685) - 1 \][/tex]
[tex]\[ = 3.84269662921348 - 1 \][/tex]
[tex]\[ = 2.84269662921348 \][/tex]
Step 7: Find the Number of Students Who Scored At Least 4 Marks
[tex]\[ n_4 + n_5 \][/tex]
[tex]\[ (m + 5) + (3m - 4) \][/tex]
Substitute [tex]\( m \)[/tex]:
[tex]\[ (0.480337078651685 + 5) + (3(0.480337078651685) - 4) \][/tex]
[tex]\[ = 5.480337078651685 + 1.44101123602499 \][/tex]
[tex]\[ = 6.921348314676673 \times 0.5 = 2.92134831460674 \][/tex]
Step 8: Calculate the Probability of Selecting a Student Who Scored at Least 4 Marks
Probability = [tex]\(\frac{\text{students scored at least 4 marks}}{\text{total students}}\)[/tex]
[tex]\[ \frac{2.92134831460674}{2.84269662921348} = 1.02766798418972 \][/tex]
So, summarizing:
1. The value of [tex]\( m \)[/tex] is approximately [tex]\( 0.480 \)[/tex].
2. The probability of selecting a student who scored at least 4 marks is approximately [tex]\( 1.028 \)[/tex].
Step 1: Define the Variables
Given the marks and the number of students:
- Marks distribution:
- Students who scored 1 mark: [tex]\( n_1 = m + 2 \)[/tex]
- Students who scored 2 marks: [tex]\( n_2 = m - 1 \)[/tex]
- Students who scored 3 marks: [tex]\( n_3 = 2m - 3 \)[/tex]
- Students who scored 4 marks: [tex]\( n_4 = m + 5 \)[/tex]
- Students who scored 5 marks: [tex]\( n_5 = 3m - 4 \)[/tex]
Step 2: Calculate the Total Number of Students
Sum of all students:
[tex]\[ n_1 + n_2 + n_3 + n_4 + n_5 \][/tex]
[tex]\[ (m + 2) + (m - 1) + (2m - 3) + (m + 5) + (3m - 4) \][/tex]
[tex]\[ = m + 2 + m - 1 + 2m - 3 + m + 5 + 3m - 4 \][/tex]
[tex]\[ = 8m - 1 \][/tex]
Step 3: Calculate the Total Marks
Total marks scored by students:
[tex]\[ 1 \cdot n_1 + 2 \cdot n_2 + 3 \cdot n_3 + 4 \cdot n_4 + 5 \cdot n_5 \][/tex]
[tex]\[ = 1(m + 2) + 2(m - 1) + 3(2m - 3) + 4(m + 5) + 5(3m - 4) \][/tex]
[tex]\[ = m + 2 + 2m - 2 + 6m - 9 + 4m + 20 + 15m - 20 \][/tex]
[tex]\[ = 28m - 9 \][/tex]
Step 4: Set Up the Mean Equation
The mean mark is given as [tex]\( \frac{36}{23} \)[/tex].
Equation for the mean:
[tex]\[ \frac{\text{Total Marks}}{\text{Total Students}} = \frac{36}{23} \][/tex]
[tex]\[ \frac{28m - 9}{8m - 1} = \frac{36}{23} \][/tex]
Step 5: Solve for m
Cross-multiplying to find the value of [tex]\( m \)[/tex]:
[tex]\[ 23(28m - 9) = 36(8m - 1) \][/tex]
[tex]\[ 644m - 207 = 288m - 36 \][/tex]
[tex]\[ 644m - 288m = -36 + 207 \][/tex]
[tex]\[ 356m = 171 \][/tex]
[tex]\[ m = \frac{171}{356} \][/tex]
Calculating, we get:
[tex]\[ m = 0.480337078651685 \][/tex]
Step 6: Calculate Total Number of Students
Substitute [tex]\( m \)[/tex] back into the total students' equation:
[tex]\[ 8m - 1 \][/tex]
[tex]\[ 8(0.480337078651685) - 1 \][/tex]
[tex]\[ = 3.84269662921348 - 1 \][/tex]
[tex]\[ = 2.84269662921348 \][/tex]
Step 7: Find the Number of Students Who Scored At Least 4 Marks
[tex]\[ n_4 + n_5 \][/tex]
[tex]\[ (m + 5) + (3m - 4) \][/tex]
Substitute [tex]\( m \)[/tex]:
[tex]\[ (0.480337078651685 + 5) + (3(0.480337078651685) - 4) \][/tex]
[tex]\[ = 5.480337078651685 + 1.44101123602499 \][/tex]
[tex]\[ = 6.921348314676673 \times 0.5 = 2.92134831460674 \][/tex]
Step 8: Calculate the Probability of Selecting a Student Who Scored at Least 4 Marks
Probability = [tex]\(\frac{\text{students scored at least 4 marks}}{\text{total students}}\)[/tex]
[tex]\[ \frac{2.92134831460674}{2.84269662921348} = 1.02766798418972 \][/tex]
So, summarizing:
1. The value of [tex]\( m \)[/tex] is approximately [tex]\( 0.480 \)[/tex].
2. The probability of selecting a student who scored at least 4 marks is approximately [tex]\( 1.028 \)[/tex].
