| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2004 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Conditional probability with Poisson |
| Difficulty | Standard +0.8 Part (i) is routine Poisson calculation. Part (ii) requires complement rule but is standard. Part (iii) is conceptually demanding—students must recognize that given a total of 5 planes in an hour, the distribution of planes between two half-hour periods follows a binomial distribution (not Poisson), requiring understanding of conditional probability and the relationship between Poisson processes and uniform distribution of events. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(P(5) = e^{-6} \times \frac{6^5}{5!} = 0.161\) | M1 A1 | For an attempted Poisson P(5) calculation, any mean; For correct answer |
| 2 | ||
| (ii) \(P(X \geq 2) = 1 - [P(0) + P(1)] = 1 - e^{-1.6}(1 + 1.6) = 0.475\) | B1 M1 A1 | For \(\mu = 1.6\), evaluated in a Poisson prob; For \(1 - P(0) - P(1)\) or \(1 - P(0) - P(1) - P(2)\); For correct answer |
| 3 | ||
| (iii) \(P(1 \text{ then } 4 | 5) = \frac{(e^{-3} \times 3) \times (e^{-3} \times \frac{3^4}{4!})}{e^{-6} \times \frac{6^5}{5!}} = 0.156\) or \(\frac{5}{32}\) | M1 M1 A1 |
| 3 |
**(i)** $P(5) = e^{-6} \times \frac{6^5}{5!} = 0.161$ | M1 A1 | For an attempted Poisson P(5) calculation, any mean; For correct answer
| | **2** | |
**(ii)** $P(X \geq 2) = 1 - [P(0) + P(1)] = 1 - e^{-1.6}(1 + 1.6) = 0.475$ | B1 M1 A1 | For $\mu = 1.6$, evaluated in a Poisson prob; For $1 - P(0) - P(1)$ or $1 - P(0) - P(1) - P(2)$; For correct answer
| | **3** | |
**(iii)** $P(1 \text{ then } 4 | 5) = \frac{(e^{-3} \times 3) \times (e^{-3} \times \frac{3^4}{4!})}{e^{-6} \times \frac{6^5}{5!}} = 0.156$ or $\frac{5}{32}$ | M1 M1 A1 | For multiplying P(1) by P(4) any (consistent) mean; For dividing by P(5) any mean; For correct answer
| | **3** | |6 At a certain airfield planes land at random times at a constant average rate of one every 10 minutes.\\
(i) Find the probability that exactly 5 planes will land in a period of one hour.\\
(ii) Find the probability that at least 2 planes will land in a period of 16 minutes.\\
(iii) Given that 5 planes landed in an hour, calculate the conditional probability that 1 plane landed in the first half hour and 4 in the second half hour.
\hfill \mbox{\textit{CAIE S2 2004 Q6 [8]}}