(a)     Find the percentage of the population that has been vaccinated.

(b)     A randomly chosen person catches the virus. Find the probability that this person has been vaccinated.

(a)

using the law of total probabilities:     (M1)

$0.1p + 0.3\left( {1 - p} \right) = 0.22$     A1

$0.1p + 0.3 - 0.3p = 0.22$

$0.2p = 0.88$

$p = \frac{{0.88}}{{0.2}} = 0.4$

$p = 40\%$ (accept $0.4$)     A1

(b)     required probability $= \frac{{0.4 \times 0.1}}{{0.22}}$     M1

$= \frac{2}{{11}}$   ($0.182$)     A1

[5 marks]

Most candidates who successfully answered this question had first drawn a tree diagram, using a symbol to denote the probability that a randomly chosen person had received the influenza virus. For those who did not draw a tree diagram, there was poor understanding of how to apply the conditional probability formula.