| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Explain or apply conditions in context |
| Difficulty | Moderate -0.8 This question tests basic understanding of Poisson distribution conditions and a straightforward calculation using the standard formula. Parts (i) and (ii) require recall of standard conditions (independence, constant rate), while part (iii) is a direct application of the Poisson probability formula with λ=2.75. No problem-solving or novel insight required—purely routine bookwork and calculation. |
| 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 |
|---|---|---|
| Answer | Marks | Guidance |
| That they don't occur regularly or to a fixed pattern, or are unpredictable | B1 | Any similar or equivalent statement, but *not* independent or equivalent. Both right and wrong: B0 |
| Total: 1 | E.g. "no pattern": expect to be right. E.g. "doesn't affect": expect to be wrong |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Dead rabbits occur independently, i.e., one occurrence does not affect the probability of another | B1 | Correct statement of principle |
| constant average rate, i.e. mean number uniform along the whole road | B1 | Correct interpretation of that principle. Context needed for any marks. SR: "Constant rate" B0, correct reason can get B1 if "average" implied |
| Total: 2 | *Not* "constant probability". One right, one wrong e.g. independent + "\(np < 5\), \(nq < 5\)": max 1. Only "Singly" stated, implied or used: max B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Po}(2.75)\) | M1 | \(\text{Po}(1650/600)\) attempted |
| \(e^{-2.75}\frac{2.75^3}{3!} = \mathbf{0.2215}\) | M1 | Correct formula, any numerical \(\lambda\) |
| A1 | Answer in range \([0.221, 0.222]\) | |
| Total: 3 | Formula required, so no formula \(\Rightarrow\) M0A0 |
## Question 2:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| That they don't occur regularly or to a fixed pattern, or are unpredictable | B1 | Any similar or equivalent statement, but *not* independent or equivalent. Both right and wrong: B0 |
| **Total: 1** | | E.g. "no pattern": expect to be right. E.g. "doesn't affect": expect to be wrong |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Dead rabbits occur independently, i.e., one occurrence does not affect the probability of another | B1 | Correct statement of principle |
| constant average rate, i.e. mean number uniform along the whole road | B1 | Correct interpretation of that principle. Context needed for any marks. **SR:** "Constant rate" B0, correct reason can get B1 if "average" implied |
| **Total: 2** | | *Not* "constant probability". One right, one wrong e.g. independent + "$np < 5$, $nq < 5$": max 1. Only "Singly" stated, implied or used: max B1 |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Po}(2.75)$ | M1 | $\text{Po}(1650/600)$ attempted |
| $e^{-2.75}\frac{2.75^3}{3!} = \mathbf{0.2215}$ | M1 | Correct formula, any numerical $\lambda$ |
| | A1 | Answer in range $[0.221, 0.222]$ |
| **Total: 3** | | Formula required, so no formula $\Rightarrow$ M0A0 |
---2 A class investigated the number of dead rabbits found along a particular stretch of road.\\
(i) The class agrees that dead rabbits occur randomly along the road. Explain what this statement means.\\
(ii) State, in this context, an assumption needed for the number of dead rabbits in a fixed length of road to be modelled by a Poisson distribution, and explain what your statement means.
Assume now that the number of dead rabbits in a fixed length of road can be well modelled by a Poisson distribution with mean 1 per 600 m of road.\\
(iii) Use an appropriate formula, showing your working, to find the probability that in a road of length 1650 m there are exactly 3 dead rabbits.
\hfill \mbox{\textit{OCR S2 2015 Q2 [6]}}