| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | October |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Consecutive non-overlapping periods |
| Difficulty | Standard +0.8 This is a comprehensive multi-part Poisson question requiring rate scaling, cumulative probability calculations, conditional probability, and independence reasoning. Part (a) requires iterative calculation to find threshold n, part (b) involves logarithms or tables, part (c) tests understanding of independence across periods, part (d) is straightforward application, and part (e) requires conditional probability with Poisson distributions. The variety of techniques and conceptual depth (especially parts c and e) elevate this above standard S2 fare, though each individual part uses established methods. |
| Spec | 5.02i 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) | \(X \sim \text{Po}(1)\) and \(P(X > n) < 0.05\) or \(P(X < n) > 0.95\) \(n = 4\) | M1 A1 (2) |
| (b) | \(Y \sim \text{Po}(0.5m)\) \(P(Y = 0) < 0.05\) or \(e^{-1m} < 0.05\) \(0.5m = 3\) \(m = 6\) | M1 A1 (2) |
| (c) | \(W \sim \text{Po}(0.5)\) \([P(W \geq 1)] = [1 - P(W = 0)]^3 = (1 - 0.6065)^3 = 0.06093\ldots\) awrt \(\mathbf{0.0609}\) | M1 M1 A1 (3) |
| (d) | \(S \sim \text{Po}(4)\) \(P(S = 4) = 0.1953\ldots\) awrt \(\mathbf{0.195}\) | M1 A1 (2) |
| (e) | \(A \sim \text{Po}(1)\) \(B \sim \text{Po}(2)\) \(P(\text{sightings in last 2 months} > \text{sightings in first 2 months} \mid 4 \text{ sightings in 4 months}) = \frac{P(A=1) \times P(A=3) + P(A=0) \times (A=4) \cdot e^{-1} \times e_0^3_0 + e^{-1} \times e_5^4_4}{\underbrace{P(B=4)}_{\text{}}} = \frac{e^{-2\times 3^4}}{e^{-2\times 2^4}} = \frac{5}{16}\) | M1 M1 A1 (3) Total 12 |
(a) | $X \sim \text{Po}(1)$ and $P(X > n) < 0.05$ or $P(X < n) > 0.95$ $n = 4$ | M1 A1 (2) | M1 for writing or using Po(1) and $P(X > n) < 0.05$ or $P(X \leq 2) = 0.9197$ or $P(X \leq 3) = 0.9810$ or $P(X > 2) = 0.0803$ or $P(X > 3) = 0.019$
(b) | $Y \sim \text{Po}(0.5m)$ $P(Y = 0) < 0.05$ or $e^{-1m} < 0.05$ $0.5m = 3$ $m = 6$ | M1 A1 (2) | M1 for $P(Y = 0) = 0.1353$ from Po(2) or $P(Y = 0) = 0.0821$ from Po(2.5) or $P(Y = 0) = 0.0498$ from Po(3)
(c) | $W \sim \text{Po}(0.5)$ $[P(W \geq 1)] = [1 - P(W = 0)]^3 = (1 - 0.6065)^3 = 0.06093\ldots$ awrt $\mathbf{0.0609}$ | M1 M1 A1 (3) | 1st M1 for $P(W \geq 1)$ and Po(0.5) / 2nd M1 for $(1 - P(W = 0))^3$ (allow $W \sim \text{Po}(\lambda)$, $\lambda > 0$)
(d) | $S \sim \text{Po}(4)$ $P(S = 4) = 0.1953\ldots$ awrt $\mathbf{0.195}$ | M1 A1 (2) | M1 for use of Po(4)
(e) | $A \sim \text{Po}(1)$ $B \sim \text{Po}(2)$ $P(\text{sightings in last 2 months} > \text{sightings in first 2 months} \mid 4 \text{ sightings in 4 months}) = \frac{P(A=1) \times P(A=3) + P(A=0) \times (A=4) \cdot e^{-1} \times e_0^3_0 + e^{-1} \times e_5^4_4}{\underbrace{P(B=4)}_{\text{}}} = \frac{e^{-2\times 3^4}}{e^{-2\times 2^4}} = \frac{5}{16}$ | M1 M1 A1 (3) Total 12 | 1st M1 for $P(A = 1) \times P(A = 3) = [0.02255...]$ $+ P(A = 0) \times P(A = 4) = [=0.0056...]$ from Po(1) / 2nd M1 for conditional probability with $P(B = 4) = [0.0902...]$ on the denom from Po(2) (M0 if num > denom) / A1 allow awrt 0.312/0.313
---7. Members of a conservation group record the number of sightings of a rare animal. The number of sightings follows a Poisson distribution with a rate of 1 every 2 months.
\begin{enumerate}[label=(\alph*)]
\item Find the smallest value of $n$ such that the probability that there are at least $n$ sightings in 2 months is less than 0.05
\item Find the smallest number of months, $m$, such that the probability of no sightings in $m$ months is less than 0.05
\item Find the probability that there is at least 1 sighting per month in each of 3 consecutive months.
\item Find the probability that the number of sightings in an 8 month period is equal to the expected number of sightings for that period.
\item Given that there were 4 sightings in a 4 month period, find the probability that there were more sightings in the last 2 months than in the first 2 months.
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\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2018 Q7 [12]}}