The Mathematics of Maybe: Introduction to Probability | Exercise 7.2

Question 2

A survey is conducted at a school where a random sample of 40 students is asked about their favourite club. The responses are are: 14 students: Science Club | 11 students: Arts Club | 9 students: Sports Club | 6 students: Debate Club Assume there are 800 students in the whole school.

(i) What is the probability that a randomly chosen student from the sample prefers the Arts Club?

(ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.

Solution

Understand the Question

Sample data ( $n = 40$ ): Science $= 14$ , Arts $= 11$ , Sports $= 9$ , Debate $= 6$ .
Total school population ( $N$ ): $800$ students.
To find the probability of a specific preference from the sample, use: $P(\text{Event}) = \dfrac{\text{Number of students with the preference}}{\text{Total students in the sample}}$
To estimate the number of students in the entire school with a preference, multiply the sample probability by the total population: $\text{Estimated count} = P(\text{Event}) \times \text{Total students in school}$

(i) What is the probability that a randomly chosen student from the sample prefers the Arts Club?

Step 1 · Calculate Probability for Arts Club

Given:

Number of students preferring Arts Club $= 11$
Total students in sample $= 40$

\begin{aligned} P(\text{Arts Club}) &= \dfrac{\text{Number of students preferring Arts Club}}{\text{Total students in sample}} \\[0.6em] &= \dfrac{11}{40} \\[0.6em] &= 0.275 \end{aligned}

Answer

(i) $\dfrac{11}{40} \text{ (or } 0.275\text{)}$

(ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.

Step 1 · Estimate Students for Sports Club

From the sample:

\begin{aligned} P(\text{Sports Club}) &= \dfrac{\text{Number of students preferring Sports Club}}{\text{Total students in sample}} \\[0.6em] &= \dfrac{9}{40} \end{aligned}

For the total school population of $800$ students:

\begin{aligned} \text{Estimated students} &= P(\text{Sports Club}) \times \text{Total students in school} \\[0.6em] &= \dfrac{9}{40} \times 800 \\[0.6em] &= 9 \times 20 \\[0.6em] &= 180 \end{aligned}

Answer

(ii) $180\text{ students}$

Common Mistakes

Wrong Denominator: Using the total school population ( $800$ ) instead of the sample size ( $40$ ) when computing the sample probability $P(\text{Arts Club})$ .
Skipping the Multiplication Step: Stopping after finding the sample probability $\dfrac{9}{40}$ without multiplying by $800$ to estimate the actual student count in the whole school.

Question 2

More questions in Exercise 7.2