| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Topic | Poisson distribution |
| Type | Single period normal approximation - large lambda direct |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution properties with standard techniques: scaling the rate parameter for different time periods, using the Poisson formula, and applying normal approximation. Part (ii) requires solving for λ using logarithms, and part (iii) is a routine normal approximation with continuity correction. All parts follow textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.02m Poisson: mean = variance = lambda5.04b Linear combinations: of normal distributions |
Total: 10 marks
**Part (i):**
$\text{Po}(4)$ **M1** Po(4) and "1 –" in tables, e.g. 0.2149 or 0.0511
$1 - P(\leqslant 6) = \text{awrt } 0.111$ **A1**
**Part (ii):**
$e^{-\lambda} = 0.6$ **M1** ($\lambda$ could be $t/180$ or $t/3$ or $20t$ depending on units)
$\lambda = -\ln 0.6$ or awrt $0.511$ **A1** soi
$t = 92$ seconds **A1** Answer 92 seconds or 1 minute 32 seconds only
**Part (iii):**
$\text{Po}(60) \approx N(60, 60)$ **M1A1** Po(60) and N(60, …); N(60, 60) soi
Standardise **M1**
Correct $\sqrt{}$ and cc **A1**
$\Phi\!\left(\dfrac{65.5 - 60}{\sqrt{60}}\right) = \Phi(0.71) = \text{awrt } 0.761$ **A1** Answer, a.r.t. 0.761. [no cc, 0.7406; wrong cc, 0.7193; 60 not $\sqrt{60}$, 0.5366]
Total: **10 marks**5 The number of calls to a car breakdown service during any one hour of the day is modelled by the distribution $\operatorname { Po } ( 20 )$.\\
(i) Find the probability that in a randomly chosen 12 -minute period there are at least 7 calls to the service.\\
(ii) Find the period of time, correct to the nearest second, for which the probability that no calls are made to the service is 0.6 .\\
(iii) Use a suitable approximation to find the probability that, in a randomly chosen 3-hour period, there are no more than 65 calls to the service.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2017 Q5 [8]}}