| Date | May 2009 | Marks available | 11 | Reference code | 09M.3sp.hl.TZ0.1 |
| Level | HL only | Paper | Paper 3 Statistics and probability | Time zone | TZ0 |
| Command term | Find | Question number | 1 | Adapted from | N/A |
Question
Ahmed and Brian live in the same house. Ahmed always walks to school and Brian always cycles to school. The times taken to travel to school may be assumed to be independent and normally distributed. The mean and the standard deviation for these times are shown in the table below.
(a) Find the probability that on a particular day Ahmed takes more than 35 minutes to walk to school.
(b) Brian cycles to school on five successive mornings. Find the probability that the total time taken is less than 70 minutes.
(c) Find the probability that, on a particular day, the time taken by Ahmed to walk to school is more than twice the time taken by Brian to cycle to school.
Markscheme
(a) \(A \sim {\text{N}}(30,{\text{ }}{3^2})\)
\({\text{P}}(A > 35) = 0.0478\) (M1)A1
[2 marks]
(b) let \(X = {B_1} + {B_2} + {B_3} + {B_4} + {B_5}\)
\({\text{E}}(X) = 5{\text{E}}(B) = 60\) A1
\({\text{Var}}(X) = 5{\text{Var}}(B) = 20\) (M1)A1
\({\text{P}}(X < 70) = 0.987\) A1
[4 marks]
(c) let \(Y = A - 2B\) (M1)
\({\text{E}}(Y) = {\text{E}}(A) - 2{\text{E}}(B) = 6\) A1
\({\text{Var}}(Y) = {\text{Var}}(A) + 4{\text{Var}}(B) = 25\) (M1)A1
\({\text{P}}(Y > 0) = 0.885\) A1
[5 marks]
Total [11 marks]
Examiners report
Most candidates were able to access this question, but weaker candidates did not always realise that parts (b) and (c) were testing different things. Part (b) proved the hardest with a number of candidates not understanding how to find the variance of the sum of variables.
