WAEC 2022 · Paper 2 · Q12

(a)
The probability that an athlete will not win any of the three races is $\frac14$ . If the athlete runs in all the races, what is the probability that the athlete will win: (i) only the second race; (ii) all the three races; (iii) only two of the races?
Separate values with commas, e.g. 3, −2
(b)
A cone with perpendicular height $24\text{ cm}$ has a volume of $1200\text{ cm}^3$ . Find the volume of a cone with the same base radius and height $84\text{ cm}$ .

Worked solution (try it first)

Read "will not win any of the races" as: the probability of losing each race is $\frac14$ , so the probability of winning each race is $\frac34$ , and the races are independent.

Only the second race: lose, win, lose: $\frac14 \times \frac34 \times \frac14 = \frac{3}{64}$ .

Only two: win-win-lose, win-lose-win or lose-win-win.
Each has probability $\frac34 \times \frac34 \times \frac14 = \frac{9}{64}$ , so together $3 \times \frac{9}{64} = \frac{27}{64}$ .

$V = \frac13\pi r^2 h$ .
With the same radius, the volume is proportional to the height: $V = 1200 \times \frac{84}{24}$
$= 1200 \times 3.5$
$= 4200\text{ cm}^3$ .