The problem provides the distribution of marks scored by students in a test. We are given the marks (1 to 5) and the number of students who scored each mark, expressed in terms of $p$. Part (a) states that the mean mark is $3\frac{5}{22}$ and asks us to find the value of $p$. Part (b) asks us to find the probability of selecting a student who scored at least 4 marks.

1. Problem Description

The problem provides the distribution of marks scored by students in a test. We are given the marks (1 to 5) and the number of students who scored each mark, expressed in terms of $p$. Part (a) states that the mean mark is $3\frac{5}{22}$ and asks us to find the value of $p$. Part (b) asks us to find the probability of selecting a student who scored at least 4 marks.

2. Solution Steps

(a) To find the value of $p$, we first need to express the mean mark in terms of $p$. The mean is calculated by summing the product of each mark and its frequency (number of students) and then dividing by the total number of students.

Total number of students $= (p+2) + (p-1) + (2p-3) + (p+4) + (3p-4) = p + 2 + p - 1 + 2p - 3 + p + 4 + 3p - 4 = 8p - 2$.

Sum of (mark $\times$ number of students):

$1(p+2) + 2(p-1) + 3(2p-3) + 4(p+4) + 5(3p-4) = p+2 + 2p-2 + 6p-9 + 4p+16 + 15p-20 = 28p - 13$.

The mean mark is $\frac{28p-13}{8p-2}$. We are given that the mean mark is $3\frac{5}{22} = \frac{3 \times 22 + 5}{22} = \frac{66+5}{22} = \frac{71}{22}$.

So, we have the equation

$\frac{28p-13}{8p-2} = \frac{71}{22}$.

Cross-multiply:

$22(28p-13) = 71(8p-2)$

$616p - 286 = 568p - 142$

$616p - 568p = 286 - 142$

$48p = 144$

$p = \frac{144}{48} = 3$

(b) Now that we have found $p=3$, we can find the number of students who scored at least 4 marks. This includes students who scored 4 and

5.

Number of students who scored 4 marks $= p+4 = 3+4 = 7$.

Number of students who scored 5 marks $= 3p-4 = 3(3)-4 = 9-4 = 5$.

Total number of students who scored at least 4 marks $= 7 + 5 = 12$.

Total number of students $= 8p - 2 = 8(3) - 2 = 24 - 2 = 22$.

Probability of selecting a student who scored at least 4 marks $= \frac{\text{Number of students who scored at least 4 marks}}{\text{Total number of students}} = \frac{12}{22} = \frac{6}{11}$.

3. Final Answer

(a) $p = 3$

(b) $\frac{6}{11}$