| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Standard +0.3 Part (i) requires a straightforward Poisson calculation with rate adjustment (3.6 per minute → 1.2 per 20 seconds) and finding P(X > 3). Part (ii) involves recognizing that a Poisson with large mean (216) can be approximated by a normal distribution, then applying continuity correction. Both are standard textbook applications with no novel insight required, making this slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x! |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((\lambda) = 3.6 \div 3 = 1.2\) | B1 | 1.2 seen |
| \(1 - e^{-1.2}\left(1 + 1.2 + \frac{1.2^2}{2} + \frac{1.2^3}{3!}\right)\) | M1 | Allow any \(\lambda\) |
| \(= 0.0338\) (3 sf) | A1 [3] | As final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(N(60 \times 3.6,\ 60 \times 3.6)\) | M1 | Stated or implied |
| \(\frac{240.5 - 216}{\sqrt{216}}\) \((= 1.667)\) | M1 | Allow with no or wrong cc (no sd/var mixes) |
| \(1 - \Phi(1.667)\) | M1 | Area consistent with their working |
| \(= 0.0478\) (3 sf) | A1 [4] | SR use of Poisson 0.0497 scores 4/4 |
## Question 3:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(\lambda) = 3.6 \div 3 = 1.2$ | B1 | 1.2 seen |
| $1 - e^{-1.2}\left(1 + 1.2 + \frac{1.2^2}{2} + \frac{1.2^3}{3!}\right)$ | M1 | Allow any $\lambda$ |
| $= 0.0338$ (3 sf) | A1 [3] | As final answer |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $N(60 \times 3.6,\ 60 \times 3.6)$ | M1 | Stated or implied |
| $\frac{240.5 - 216}{\sqrt{216}}$ $(= 1.667)$ | M1 | Allow with no or wrong cc (no sd/var mixes) |
| $1 - \Phi(1.667)$ | M1 | Area consistent with their working |
| $= 0.0478$ (3 sf) | A1 [4] | SR use of Poisson 0.0497 scores 4/4 |
---3 Particles are emitted randomly from a radioactive substance at a constant average rate of 3.6 per minute. Find the probability that\\
(i) more than 3 particles are emitted during a 20 -second period,\\
(ii) more than 240 particles are emitted during a 1-hour period.
\hfill \mbox{\textit{CAIE S2 2016 Q3 [7]}}