Question
In a given week, the number of students in a particular primary school that were absent due to headlice (H), influenza (I), and/or chickenpox (C) were recorded as follows. The primary school has 500 students.
- 35 students had headlice only
- 20 students had influenza only
- 5 students had chickenpox only
- 4 students had headlice and influenza but not chickenpox
- 2 students had headlice and chickenpox but not influenza
- 3 students had influenza and chickenpox but not headlice
- 1 student had headlice, influenza, and chickenpox
(a) Draw a Venn diagram to represent this information.
(b) Calculate the number of students who did not have headlice or influenza or chickenpox.
A student is chosen at random from all the students in the school.
(c) Find the probability that this student has:
(i) Headlice.
(ii) Influenza given that the student has headlice.
Diego is a teacher in the school. He believes that the number of students, \( n \), who have had influenza during the first \( t \) days of the school year, can be modelled by the function:
\[ n(t) = 250 – 240(2)^{kt}, \quad k \in \mathbb{R}. \]
(d) Use Diego’s model to calculate the number of students who started the school year with influenza.
It is known that 130 students have had influenza during the first 10 days of the school year.
(e) Find the value of \( k \).
(f) Using this model, calculate how many days it will take for 200 students to have had influenza since the start of the school year.
By the last day of the school year, it is known that 300 students have had influenza.
(g) Comment on the appropriateness of Diego’s model.
▶️Answer/Explanation
Detailed solution
(a) Venn Diagram:
The Venn diagram is drawn as follows:
- Headlice (H): 35 (only), 4 (H and I), 2 (H and C), 1 (H, I, and C).
- Influenza (I): 20 (only), 4 (H and I), 3 (I and C), 1 (H, I, and C).
- Chickenpox (C): 5 (only), 2 (H and C), 3 (I and C), 1 (H, I, and C).
The Venn diagram is shown below:
(b) Number of Students Without Any Condition:
First, calculate the total number of students with at least one condition:
\[ 35 (\text{H only}) + 4 (\text{H and I}) + 20 (\text{I only}) + 3 (\text{I and C}) + 1 (\text{H, I, and C}) + 2 (\text{H and C}) + 5 (\text{C only}) = 70 \]
Thus, the number of students without any condition is:
\[ 500 – 70 = 430 \]
So, 430 students did not have headlice, influenza, or chickenpox.
(c) Probabilities:
(i) Probability that a student has headlice:
The total number of students with headlice is:
\[ 35 (\text{H only}) + 4 (\text{H and I}) + 2 (\text{H and C}) + 1 (\text{H, I, and C}) = 42 \]
Thus, the probability is:
\[ P(\text{Headlice}) = \frac{42}{500} = 0.084 \, \text{or} \, 8.4\% \]
(ii) Probability that a student has influenza given that they have headlice:
The number of students with both headlice and influenza is:
\[ 4 (\text{H and I}) + 1 (\text{H, I, and C}) = 5 \]
Thus, the conditional probability is:
\[ P(\text{Influenza} \mid \text{Headlice}) = \frac{5}{42} \approx 0.119 \, \text{or} \, 11.9\% \]
(d) Number of Students with Influenza at Start of School Year:
At \( t = 0 \), substitute into Diego’s model:
\[ n(0) = 250 – 240(2)^{k \cdot 0} = 250 – 240(1) = 10 \]
So, 10 students started the school year with influenza.
(e) Value of \( k \):
Given that \( n(10) = 130 \), substitute into the model:
\[ 130 = 250 – 240(2)^{10k} \]
Solve for \( k \):
\[ 240(2)^{10k} = 250 – 130 = 120 \]
\[ (2)^{10k} = \frac{120}{240} = 0.5 \]
Take the logarithm base 2 of both sides:
\[ 10k = \log_2(0.5) = -1 \]
\[ k = -\frac{1}{10} = -0.1 \]
So, \( k = -0.1 \).
(f) Days for 200 Students to Have Influenza:
We need to find \( t \) such that \( n(t) = 200 \):
\[ 200 = 250 – 240(2)^{-0.1t} \]
Solve for \( t \):
\[ 240(2)^{-0.1t} = 250 – 200 = 50 \]
\[ (2)^{-0.1t} = \frac{50}{240} = \frac{5}{24} \]
Take the logarithm base 2 of both sides:
\[ -0.1t = \log_2\left(\frac{5}{24}\right) \]
Using a calculator:
\[ \log_2\left(\frac{5}{24}\right) \approx -2.263 \]
\[ t = \frac{-2.263}{-0.1} \approx 22.63 \, \text{days} \]
So, it will take approximately 22.6 days for 200 students to have had influenza.
(g) Appropriateness of Diego’s Model:
Diego’s model predicts that the number of students who have had influenza will approach 250 as \( t \) increases, but it will never exceed 250. However, by the last day of the school year, 300 students have had influenza, which is greater than 250. Therefore, Diego’s model is not appropriate because it does not account for the observed data.
……………………………Markscheme……………………………….
(a) Venn diagram provided.
(b) Number of students without any condition: 430.
(c) (i) Probability of headlice: 8.4%.
(ii) Probability of influenza given headlice: 11.9%.
(d) Number of students with influenza at start: 10.
(e) Value of \( k \): -0.1.
(f) Days for 200 students to have influenza: 22.6 days.
(g) Diego’s model is not appropriate because it does not predict the observed number of 300 students.