| Date | November 2015 | Marks available | 9 | Reference code | 15N.3sp.hl.TZ0.3 |
| Level | HL only | Paper | Paper 3 Statistics and probability | Time zone | TZ0 |
| Command term | Calculate | Question number | 3 | Adapted from | N/A |
Question
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.
Markscheme
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]
