| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | March |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson approximation justification or comparison |
| Difficulty | Moderate -0.3 This is a straightforward application of Poisson approximation to binomial with standard conditions (large n, small p, moderate λ). Part (a) requires calculating P(X≥3) = 1 - P(X≤2) using λ = np = 0.6, which is routine. Part (b) asks for standard justification criteria (n large, p small, np moderate). No conceptual difficulty or novel insight required, just textbook application, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x! |
| Answer | Marks | Guidance |
|---|---|---|
| \([\lambda =]\ 0.6\) | B1 | Mean = 0.6 seen |
| \(1 - e^{-0.6}\left(1 + 0.6 + \frac{0.6^2}{2}\right)\) or \(1 - e^{-0.6}(1 + 0.6 + 0.18)\) or \(1 - (0.5488 + 0.3293 + 0.09879)\) | M1 | Any \(\lambda\). Allow one end error. Must see expression. Accept correct \(\Sigma\) notation. |
| \(= 0.0231\) | A1 | SC 0.0231 with no working scores B1 (could be implied). SC use of binomial scores M1A1 for 0.0231. |
| Answer | Marks | Guidance |
|---|---|---|
| \(6000 > 50\) and either \(np = 0.6 < 5\) or \(\frac{1}{10000} < 0.1\) | B1 | Must state values of \(n\) and either \(np\) or \(p\). Note: '\(n\) large, \(p\) small' is insufficient. |
## Question 3(a):
| $[\lambda =]\ 0.6$ | B1 | Mean = 0.6 seen |
|---|---|---|
| $1 - e^{-0.6}\left(1 + 0.6 + \frac{0.6^2}{2}\right)$ or $1 - e^{-0.6}(1 + 0.6 + 0.18)$ or $1 - (0.5488 + 0.3293 + 0.09879)$ | M1 | Any $\lambda$. Allow one end error. Must see expression. Accept correct $\Sigma$ notation. |
| $= 0.0231$ | A1 | SC 0.0231 with no working scores B1 (could be implied). SC use of binomial scores M1A1 for 0.0231. |
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## Question 3(b):
| $6000 > 50$ and either $np = 0.6 < 5$ or $\frac{1}{10000} < 0.1$ | B1 | Must state values of $n$ and either $np$ or $p$. Note: '$n$ large, $p$ small' is insufficient. |
---3 In a certain lottery, on average 1 in every 10000 tickets is a prize-winning ticket. An agent sells 6000 tickets.
\begin{enumerate}[label=(\alph*)]
\item Use a suitable approximating distribution to find the probability that at least 3 of the tickets sold by the agent are prize-winning tickets.
\item Justify the use of your approximating distribution in this context.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q3 [4]}}