Two students are selected at random from a large school with equal numbers of boys and girls. The boys’ heights are normally distributed with mean $178$ cm and standard deviation $5.2$ cm, and the girls’ heights are normally distributed with mean $169$ cm and standard deviation $5.4$ cm.

Calculate the probability that the taller of the two students selected is a boy.

let $X$ denote boys’ height and $Y$ denote girls’ height

if $BB,{\text{ P(taller is boy)}} = 1$     (A1)

if $GG,{\text{ P(taller is boy)}} = 0$     (A1)

if $BG$ or $GB$:

consider $X - Y$     (M1)

$E(X - Y) = 178 - 169 = 9$     A1

${\text{Var}}(X - Y) = {5.2^2} + {5.4^2}\;\;\;( = 56.2)$     (M1)A1

${\text{P}}(X - Y > 0) = 0.885$     A1

answer is $\frac{1}{4} \times 1 + \frac{1}{2} \times 0.885 = 0.693$     (M1)A1

[9 marks]