2 Before starting the scholastic year, $$ \frac{5}{7} $$ of the students in a school, have had a flu vaccine. If a student has had the vaccine, the probability of catching the flu is $$ \frac{1}{35} $$. If a student has NOT had the vaccine, the probability of catching the flu is $$ \frac{9}{14} $$. (i) Draw a tree diagram to show the given data. (ii) Calculate the probability that a student, chosen at random from this school, does NOT catch the flu. (iii) It is expected that around 150 students from this school will catch the flu. How many students attend this school?

Question

UpStudy AI · Accepted Answer

Let's tackle each part of the problem step by step.

### **Given:**
- **Total Students:** Let’s denote the total number of students as $$ N $$.
- **Vaccinated Students:** $$ \frac{5}{7} $$ of the students have received the flu vaccine.
- **Unvaccinated Students:** $$ \frac{2}{7} $$ of the students have **not** received the flu vaccine.
- **Probability of Catching the Flu:**
  - **If Vaccinated:** $$ \frac{1}{35} $$
  - **If Not Vaccinated:** $$ \frac{9}{14} $$

---

### **(i) Tree Diagram Representation**

While I can't draw a tree diagram here, I can describe how it would look:

1. **First Level (Vaccination Status):**
   - **Branch 1:** **Vaccinated** with probability $$ \frac{5}{7} $$.
   - **Branch 2:** **Not Vaccinated** with probability $$ \frac{2}{7} $$.

2. **Second Level (Flu Status):**
   - **From Vaccinated Branch:**
     - **Catch Flu:** Probability $$ \frac{1}{35} $$.
     - **Do Not Catch Flu:** Probability $$ \frac{34}{35} $$ (since $$ 1 - \frac{1}{35} = \frac{34}{35} $$).
   
   - **From Not Vaccinated Branch:**
     - **Catch Flu:** Probability $$ \frac{9}{14} $$.
     - **Do Not Catch Flu:** Probability $$ \frac{5}{14} $$ (since $$ 1 - \frac{9}{14} = \frac{5}{14} $$).

The tree diagram visually represents these branches and probabilities, showing all possible outcomes.

---

### **(ii) Probability of a Student **Not** Catching the Flu**

To find the probability that a randomly chosen student does **not** catch the flu, we'll use the **Total Probability Theorem**.

$$
P(\text{Not Flu}) = P(\text{Vaccinated}) \times P(\text{Not Flu} \mid \text{Vaccinated}) + P(\text{Not Vaccinated}) \times P(\text{Not Flu} \mid \text{Not Vaccinated})
$$

Plugging in the values:

$$
P(\text{Not Flu}) = \left( \frac{5}{7} \right) \times \left( \frac{34}{35} \right) + \left( \frac{2}{7} \right) \times \left( \frac{5}{14} \right)
$$

$$
P(\text{Not Flu}) = \frac{170}{245} + \frac{10}{98} = \frac{34}{49} + \frac{5}{49} = \frac{39}{49}
$$

$$
P(\text{Not Flu}) = \frac{39}{49} \approx 0.7959 \text{ or } 79.59\%
$$

**So, the probability that a randomly chosen student does NOT catch the flu is $$ \frac{39}{49} $$ or approximately 79.59%.**

---

### **(iii) Calculating the Total Number of Students**

We’re told that **150 students** are expected to catch the flu. Let’s find the total number of students $$ N $$.

First, determine the probability of catching the flu:

$$
P(\text{Flu}) = 1 - P(\text{Not Flu}) = 1 - \frac{39}{49} = \frac{10}{49}
$$

The expected number of students catching the flu is:

$$
\text{Expected Flu Cases} = P(\text{Flu}) \times N
$$

Given that:

$$
150 = \frac{10}{49} \times N
$$

Solving for $$ N $$:

$$
N = \frac{150 \times 49}{10} = \frac{7350}{10} = 735
$$

**Therefore, approximately 735 students attend the school.**
**(i) Tree Diagram:** - **First Level:** Vaccinated ($$ \frac{5}{7} $$) and Not Vaccinated ($$ \frac{2}{7} $$). - **Second Level:** For each, branches for Catching Flu and Not Catching Flu with respective probabilities. **(ii) Probability of Not Catching the Flu:** $$ \frac{39}{49} \text{ or } 79.59\% $$ **(iii) Total Number of Students:** 735 students attend the school.

Upstudy AI Solution

Answer

Solution

Given:

(i) Tree Diagram Representation

(ii) Probability of a Student Not Catching the Flu

(iii) Calculating the Total Number of Students

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