2024 H2 Maths Paper 2 Q6 — Worked Solution

Part (a)

1 mark

The 65 respondents form a sample, not the population.

Explanation: the population here is all 90 members of the club (27 Youth $\text{+}$ 45 Adult $\text{+}$ 18 Senior), since the questionnaire was sent to every member. Only 65 of them replied, so these 65 are part — not all — of that group: a self-selected sample of the population, not the population itself.

Part (b)

1 mark

Choosing a random sample of members for the committee gives every member an equal chance of selection, so the committee is not biased by the secretary favouring particular members — it gives a fairer, more representative selection of opinion.

Part (c)

4 marks

Approach

let $y$ , $a$ , $s$ be the numbers of Youth, Adult and Senior members on the committee, so $y + a + s = 6$ with $y \geq 1$ , $a \geq 1$ , $s \geq 1$ , $a > y$ and $a > s$ . Since $a$ is a strict majority-holder among only 6 members split three ways, we can simply list the few triples satisfying these constraints, rather than set up a general formula.

Listing all triples with $y + a + s = 6$ , each at least $1$ , and $a > y$ , $a > s$ :

Case 1: $y = 1,\ a = 4,\ s = 1$

Case 2: $y = 1,\ a = 3,\ s = 2$

Case 3: $y = 2,\ a = 3,\ s = 1$

(every other combination fails $a > y$ or $a > s$ ).

Choosing each committee from the population of 27 Youth, 45 Adult and 18 Senior members,

\text{Case 1: }\binom{27}{1}\binom{45}{4}\binom{18}{1} = 27 \times 148\, 995 \times 18 = 72\, 411\, 570

\text{Case 2: }\binom{27}{1}\binom{45}{3}\binom{18}{2} = 27 \times 14\, 190 \times 153 = 58\, 618\, 890

\text{Case 3: }\binom{27}{2}\binom{45}{3}\binom{18}{1} = 351 \times 14\, 190 \times 18 = 89\, 652\, 420

Adding the three cases, the total number of committees satisfying all the stated constraints is

72\, 411\, 570 + 58\, 618\, 890 + 89\, 652\, 420 = 220\, 682\, 880

Of these, only Case 3 has $y > s$ (here $2 > 1$ ); Case 1 has $y = s$ and Case 2 has $y < s$ , so only Case 3 is favourable.

P(Y > S) = \frac{89\, 652\, 420}{220\, 682\, 880}

= \frac{13}{32}

To check your answer:

\frac{89\, 652\, 420}{220\, 682\, 880}

simplifies exactly to

\frac{13}{32} = 0.40625

✓

$\therefore$ $P(Y > S) = \frac{13}{32} = 0.406$ (3 s.f.).

2024 H2 Maths Paper 2, Question 6

Part (a)

Part (b)

Part (c)

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