WAEC 2013 · Paper 2 · Q14

(a)
A bag contains 5 blue, 4 green and 3 yellow balls. All the balls are identical except for colour. Three balls are drawn at random without replacement. Find the probability that: (i) all three balls have the same colour; (ii) exactly two balls have the same colour.
Separate values with commas, e.g. 3, −2
(b)

Physics 6 5 4 3 2 7 1

Chemistry 7 6 2 4 1 5 3

The table shows the ranks of the marks scored by 7 candidates in Physics and Chemistry tests. Calculate the Spearman's rank correlation coefficient (4 d.p.).

Physics	6	5	4	3	2	7	1
Chemistry	7	6	2	4	1	5	3

Worked solution (try it first)

All the same colour: $\binom53 + \binom43 + \binom33 = 10 + 4 + 1$
$= 15$ , so $P = \frac{15}{220} = \frac{3}{44}$ .

Exactly two the same: two of one colour and one of another: $\binom52 \times 7 + \binom42 \times 8 + \binom32 \times 9 = 70 + 48 + 27$
$= 145$ .
So $P = \dfrac{145}{220} = \dfrac{29}{44}$ .

The table already gives ranks.
$d$ : $-1, -1, 2, -1, 1, 2, -2$ , so $\sum d^2 = 16$ .
$\rho = 1 - \dfrac{6 \times 16}{7 \times 48}$
$= 1 - \dfrac{96}{336}$
$= \dfrac57$
$\approx 0.7143$ .