| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State Poisson approximation with justification |
| Difficulty | Moderate -0.8 This is a straightforward application of the standard Poisson approximation to the binomial with large n and small p. Part (a) requires calculating λ=np=4 and finding P(X=2,3,4) from Po(4), which is routine. Part (b) asks for textbook justification (n large, p small, np moderate). No problem-solving or insight required—pure recall and standard procedure. |
| Spec | 2.04c Calculate binomial probabilities5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x! |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([\lambda =]\ 4\) | B1 | |
| \(e^{-4}\left(\frac{4^2}{2!} + \frac{4^3}{3!} + \frac{4^4}{4!}\right)\) or \(e^{-4}(8 + 10.67 + 10.67)\) or \(0.1465 + 0.19537 + 0.19537\) | M1 | Allow one end error. Any \(\lambda\). Expression must be seen. |
| \(= 0.537\) (3sf) | A1 | SC B1 B1 for unsupported correct answer. SC B2 for use of Binomial leading to 0.537. Note: use of normal could score B1 only for mean \(= 4\). |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(n = 4000 > 50\) and either \(np = 4 < 5\) or \(p = 0.001 < 0.1\) | B1 | Explicit values seen. |
| 1 |
## Question 1:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[\lambda =]\ 4$ | B1 | |
| $e^{-4}\left(\frac{4^2}{2!} + \frac{4^3}{3!} + \frac{4^4}{4!}\right)$ or $e^{-4}(8 + 10.67 + 10.67)$ or $0.1465 + 0.19537 + 0.19537$ | M1 | Allow one end error. Any $\lambda$. Expression must be seen. |
| $= 0.537$ (3sf) | A1 | SC B1 B1 for unsupported correct answer. SC B2 for use of Binomial leading to 0.537. Note: use of normal could score B1 only for mean $= 4$. |
| | **3** | |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $n = 4000 > 50$ and either $np = 4 < 5$ or $p = 0.001 < 0.1$ | B1 | Explicit values seen. |
| | **1** | |
---1 The random variable $X$ has the distribution $\mathrm { B } ( 4000,0.001 )$.
\begin{enumerate}[label=(\alph*)]
\item Use a suitable approximating distribution to find $\mathrm { P } ( 2 \leqslant X < 5 )$.
\item Justify your approximating distribution in this case.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q1 [4]}}