| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Moderate -0.8 This is a straightforward application of the Poisson distribution requiring only direct substitution into the formula or calculator use. All three parts involve standard probability calculations (exact value, cumulative tail, cumulative from below) with clearly stated parameters, requiring no problem-solving insight or multi-step reasoning beyond identifying λ for each scenario. |
| 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/Working | Mark | Guidance |
| \(e^{-\frac{10}{3}} \times \frac{\left(\frac{10}{3}\right)^2}{2}\) | M1 | \(P(2)\), allow any \(\lambda\) |
| \(= 0.198\) (3 sf) | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1 - e^{-2}\!\left(1 + 2 + \frac{2^2}{2}\right)\) | M1 M1 | M1 allow any \(\lambda\) and/or 1 end error; Correct expression, correct \(\lambda\) |
| \(= 0.323\) (3 sf) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(N\!\left(\frac{200}{3},\, \frac{200}{3}\right)\) | M1 | seen or implied |
| \(\frac{49.5 - \frac{200}{3}}{\sqrt{\frac{200}{3}}}\ (= -2.102)\) | M1 | For standardising, allow either wrong or no cc. No sd/var mix |
| \(\Phi(-2.102) = 1 - \Phi(2.102)\) \(= 0.0178\) (3 sf) | M1 A1 [4] | For finding area consistent with their working |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $e^{-\frac{10}{3}} \times \frac{\left(\frac{10}{3}\right)^2}{2}$ | M1 | $P(2)$, allow any $\lambda$ |
| $= 0.198$ (3 sf) | A1 [2] | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1 - e^{-2}\!\left(1 + 2 + \frac{2^2}{2}\right)$ | M1 M1 | M1 allow any $\lambda$ and/or 1 end error; Correct expression, correct $\lambda$ |
| $= 0.323$ (3 sf) | A1 [3] | |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $N\!\left(\frac{200}{3},\, \frac{200}{3}\right)$ | M1 | seen or implied |
| $\frac{49.5 - \frac{200}{3}}{\sqrt{\frac{200}{3}}}\ (= -2.102)$ | M1 | For standardising, allow either wrong or no cc. No sd/var mix |
| $\Phi(-2.102) = 1 - \Phi(2.102)$ $= 0.0178$ (3 sf) | M1 A1 [4] | For finding area consistent with their working |6 A publishing firm has found that errors in the first draft of a new book occur at random and that, on average, there is 1 error in every 3 pages of a first draft. Find the probability that in a particular first draft there are\\
(i) exactly 2 errors in 10 pages,\\
(ii) at least 3 errors in 6 pages,\\
(iii) fewer than 50 errors in 200 pages.
\hfill \mbox{\textit{CAIE S2 2015 Q6 [9]}}