S2 January 2006 Q2
2. Accidents on a particular stretch of motorway occur at an average rate of 1.5 per week.
(a) Write down a suitable model to represent the number of accidents per week on this stretch of motorway. (1)
Find the probability that
(b) there will be 2 accidents in the same week, (2)
(c) there is at least one accident per week for 3 consecutive weeks, (3)
(d) there are more than 4 accidents in a 2 week period. (2)
| Scheme | Marks |
|---|---|
| Let \(X\) be the random variable the no. of accidents per week \(X \sim \mathrm{Po}(1.5)\) | B1 |
| (1) |
Notes
B1 need poisson and \(\lambda\); must be in part (a)
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X = 2) = \dfrac{\mathrm{e}^{-1.5}1.5^2}{2}\) | M1 |
| \(= 0.2510\) | A1 |
| (2) |
Notes
M1 \(\dfrac{\mathrm{e}^{\mu}\mu^2}{2}\) or \(\mathrm{P}(X \leqslant 2) - \mathrm{P}(X \leqslant 1)\)
A1 awrt 0.251
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \geqslant 1) = 1 - \mathrm{P}(X = 0) = 1 - \mathrm{e}^{-1.5}\) \(= 0.7769\) | B1 |
| P(at least 1 accident per week for 3 weeks) \(= 0.7769^3\) | M1 |
| \(= 0.4689\) | A1 |
| (3) |
Notes
B1 correct exp awrt 0.777
M1 \((p)^3\)
A1 awrt 0.469
c) The 0.7769 may be implied
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{Po}(3)\) | B1 |
| \(\mathrm{P}(X \gt 4) = 1 - \mathrm{P}(X \leqslant 4)\) | M1 |
| \(= 0.1847\) | A1 |
| (3) | |
| (9 marks) |
Notes
B1 may be implied
A1 awrt 0.1847
NB the question paper prints (2) for this part but the mark scheme awards 3 marks; with 3 marks here the paper totals 75.