| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Multiple independent time periods |
| Difficulty | Standard +0.8 This is a multi-part S2 question requiring standard Poisson calculations (parts a-c), conceptual understanding of why different approaches yield different answers (part d), and normal approximation to Poisson for large λ (part e). While it tests multiple techniques and requires careful thinking about the difference between 'total < 2' and 'no more than 1', each individual step uses routine S2 methods without requiring novel insight or complex problem-solving. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) \(e^{-0.8} = 0.449\) | B1 B1 | |
| (ii) \(0.8e^{-0.8} = 0.359\) | ||
| (b) \(P(0) + P(1) = 0.449^{10} + 10 \times 0.449^9 \times 0.359 = 0.002996\) | M1 M1 A1 A1 | |
| (c) No. in 10 patches \(\sim \text{Po}(8)\); then \(P(X < 2) = P(X \leq 1) = 0.0030\) | B1 M1 A1 | |
| (d) Good agreement, but Poisson is easier to calculate | B1 B1 | |
| (e) In 1 m², expect 8000 daisies, so use \(\text{Po}(8000) \approx N(8000, 8000)\) | M1 A1 | |
| \(P(X > 8100.5) = P(Z > 100.5/89.44) = P(Z > 1.12) = 0.131\) | M1 A1 M1 A1 | Total: 17 marks |
(a) (i) $e^{-0.8} = 0.449$ | B1 B1 |
(ii) $0.8e^{-0.8} = 0.359$ | |
(b) $P(0) + P(1) = 0.449^{10} + 10 \times 0.449^9 \times 0.359 = 0.002996$ | M1 M1 A1 A1 |
(c) No. in 10 patches $\sim \text{Po}(8)$; then $P(X < 2) = P(X \leq 1) = 0.0030$ | B1 M1 A1 |
(d) Good agreement, but Poisson is easier to calculate | B1 B1 |
(e) In 1 m², expect 8000 daisies, so use $\text{Po}(8000) \approx N(8000, 8000)$ | M1 A1 |
$P(X > 8100.5) = P(Z > 100.5/89.44) = P(Z > 1.12) = 0.131$ | M1 A1 M1 A1 | Total: 17 marks7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per $\mathrm { cm } ^ { 2 }$. One particular patch, of area $1 \mathrm {~cm} ^ { 2 }$, is selected at random.
Assuming that the number of daisies per $\mathrm { cm } ^ { 2 }$ has a Poisson distribution,
\begin{enumerate}[label=(\alph*)]
\item find the probability that the chosen patch contains
\begin{enumerate}[label=(\roman*)]
\item no daisies,
\item one daisy.
Ten such patches are chosen. Using your answers to part (a),
\end{enumerate}\item find the probability that the total number of daisies is less than two.
\item By considering the distribution of daisies over patches of $10 \mathrm {~cm} ^ { 2 }$, use the Poisson distribution to find the probability that a particular area of $10 \mathrm {~cm} ^ { 2 }$ contains no more than one daisy.
\item Compare your answers to parts (b) and (c).
\item Use a suitable approximation to find the probability that a patch of area $1 \mathrm {~m} ^ { 2 }$ contains more than 8100 daisies.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q7 [17]}}