WAEC 2018 · Paper 2 · Q13

(a)
At the end of full time, a football match between teams $A$ and $B$ was goalless. The winning team had to be decided through penalty kicks. Each team had to take five kicks. The probability that a kick by team $A$ will result in a goal was 0.8 and for team $B$ the probability was 0.3. Find, correct to three decimal places, the probability that the final scores would be: (i) 4 – 2 in favour of team $B$ ; (ii) 5 – 0 in favour of team $A$ .
Separate values with commas, e.g. 3, −2
(b)
Three students are to be selected from 4 boys and 5 girls to represent their school in a Mathematics quiz. (i) In how many ways can the three students be selected? (ii) What is the probability that more boys will be selected than girls?
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

4–2 to $B$ : $B$ scores 4 of 5 and $A$ scores 2 of 5.
$P(B \text{ scores } 4) = {}^5C_4(0.3)^4(0.7)$
$= 0.02835$ and $P(A \text{ scores } 2) = {}^5C_2(0.8)^2(0.2)^3$
$= 0.0512$ .
Multiply (the teams are independent): $0.02835 \times 0.0512 \approx 0.00145$ , which is $0.001$ to three decimal places.

5–0 to $A$ : $A$ scores all 5 and $B$ misses all 5: $0.8^5 \times 0.7^5 = 0.32768 \times 0.16807$
$\approx 0.055$ .

More boys: 3 boys, or 2 boys and 1 girl: $\dfrac{{}^4C_3 + {}^4C_2 \times {}^5C_1}{84} = \dfrac{4 + 30}{84}$
$= \dfrac{17}{42}$
$\approx 0.405$ .