| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single time period probability |
| Difficulty | Moderate -0.8 This is a straightforward Poisson distribution question requiring only recognition of the appropriate model (part a), basic conceptual understanding of the parameter (part b), and direct application of standard Poisson probability formulas (parts c-d). All calculations are routine with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure and cumulative probability calculation. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda |
| Answer | Marks |
|---|---|
| (a) Poisson (with \(\lambda = 4.2\)) | B1 |
| (b)(i) e.g. may be more or less species that like nuts | B1 |
| (b)(ii) e.g. will last longer so may get more species visiting | B1 |
| (c) Let \(X =\) no. of species that visit \(\therefore X \sim \text{Po}(4.2)\); \(P(X = 6) = \frac{e^{-4.2} \times 4.2^6}{6!} = 0.1143\) (4sf) | M1 A1 |
| (d) \(P(X > 2) = 1 - P(X \leq 2) = 1 - e^{-4.2}(1 + 4.2 + \frac{4.2^2}{2}) = 1 - 0.2102 = 0.7898\) (4sf) | M1 M1 A1 |
(a) Poisson (with $\lambda = 4.2$) | B1 |
(b)(i) e.g. may be more or less species that like nuts | B1 |
(b)(ii) e.g. will last longer so may get more species visiting | B1 |
(c) Let $X =$ no. of species that visit $\therefore X \sim \text{Po}(4.2)$; $P(X = 6) = \frac{e^{-4.2} \times 4.2^6}{6!} = 0.1143$ (4sf) | M1 A1 |
(d) $P(X > 2) = 1 - P(X \leq 2) = 1 - e^{-4.2}(1 + 4.2 + \frac{4.2^2}{2}) = 1 - 0.2102 = 0.7898$ (4sf) | M1 M1 A1 |
---An ornithologist believes that on average 4.2 different species of bird will visit a bird table in a rural garden when 50 g of breadcrumbs are spread on it.
\begin{enumerate}[label=(\alph*)]
\item Suggest a suitable distribution for modelling the number of species that visit a bird table meeting these criteria. [1 mark]
\item Explain why the parameter used with this model may need to be changed if
\begin{enumerate}[label=(\roman*)]
\item 50 g of nuts are used instead of breadcrumbs,
\item 100g of breadcrumbs are used.
\end{enumerate}
[2 marks]
\end{enumerate}
A bird table in a rural garden has 50 g of breadcrumbs spread on it.
Find the probability that
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item exactly 6 different species visit the table, [2 marks]
\item more than 2 different species visit the table. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q2 [9]}}