WAEC 2013 · Paper 2 · Q13

(a)
A committee of 6 is to be selected at random from a group of 7 boys and 4 girls. (i) In how many ways can this be done if there are no restrictions? (ii) Find the probability that the committee contains at least two girls.
Separate values with commas, e.g. 3, −2
(b)
Eight students scored the following marks in theory and practical tests in Chemistry.

Students A B C D E F G H

Score in Theory 6 7 5.5 4 8 9 5 3

Score in Practical 9 7 4 3 6 8 6.5 5

Calculate Spearman's rank correlation coefficient between scores in the two tests.

Students	A	B	C	D	E	F	G	H
Score in Theory	6	7	5.5	4	8	9	5	3
Score in Practical	9	7	4	3	6	8	6.5	5

Worked solution (try it first)

At least two girls means 2, 3 or 4 girls, since there are only 4 girls.
2 girls and 4 boys: ${}^4C_2 \times {}^7C_4 = 6 \times 35$
$= 210$ .
3 girls and 3 boys: ${}^4C_3 \times {}^7C_3 = 4 \times 35$
$= 140$ .
4 girls and 2 boys: ${}^4C_4 \times {}^7C_2 = 1 \times 21 = 21$ .
Add the cases and divide by 462: $\dfrac{371}{462}$ , which simplifies to $\dfrac{53}{66} \approx 0.80$ .

Rank each test from the highest score (rank 1).
Theory, A to H: 4, 3, 5, 7, 2, 1, 6, 8.
Practical, A to H: 1, 3, 7, 8, 5, 2, 4, 6.
Differences $d$ : 3, 0, $-2$ , $-1$ , $-3$ , $-1$ , 2, 2, so $\sum d^2 = 9 + 0 + 4 + 1 + 9 + 1 + 4 + 4$
$= 32$ .
Use $r = 1 - \dfrac{6\sum d^2}{n(n^2 - 1)}$ with $n = 8$ : $r = 1 - \dfrac{6 \times 32}{8 \times 63}$ .
Work it out: $1 - \dfrac{192}{504} \approx 0.62$ .
Spearman's rank correlation coefficient is about 0.62.