| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State Poisson approximation with justification |
| Difficulty | Moderate -0.8 This is a standard textbook application of Poisson approximation to binomial with clear parameters (n=40000, p=0.0001 giving λ=4). Parts (i) and (ii) require only stating distributions and recalling the approximation conditions (large n, small p), while part (iii) involves a routine cumulative probability calculation. No problem-solving insight needed, just direct application of learned procedures. |
| Spec | 2.04b Binomial distribution: as model B(n,p)5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(B(40000, 0.0001)\) | B1 [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(Po(4)\) | B1*B1*dep | B1 for Po. B1 for 4 |
| \(n = 40000 > 50\), \(np = 4 < 5\) | B1 [3] | Accept 40000 large and 0.0001 small |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1 - (P(X \leq 3))\) or \(e^{-4}(1 + 4 + \frac{4^2}{2} + \frac{4^3}{3!})\) | M1 | Allow one end error (any \(\lambda\)) |
| \(1 - e^{-4}(1 + 4 + \frac{4^2}{2} + \frac{4^3}{3!})\) | M1 | Expression of correct form (any \(\lambda\)), no end errors |
| \(= 0.567\) or \(0.566\) | A1 [3] | OR Use of normal scores M1, standardising M1, standardising with correct cc A1ft, (ii) 0.599. Award A mark only if normal given in (ii); OR Binomial M1 expression of correct form allow end error, M1 correct form no end error, A1ft 0.567 or 0.566. Award A mark only if Bin given in (ii). NB Part (iii) must be Poisson or ft from (ii) for A mark to be awarded. SR If no answer given in (ii) allow BOD for A marks. |
## Question 3:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $B(40000, 0.0001)$ | B1 [1] | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $Po(4)$ | B1*B1*dep | B1 for Po. B1 for 4 |
| $n = 40000 > 50$, $np = 4 < 5$ | B1 [3] | Accept 40000 large and 0.0001 small |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1 - (P(X \leq 3))$ or $e^{-4}(1 + 4 + \frac{4^2}{2} + \frac{4^3}{3!})$ | M1 | Allow one end error (any $\lambda$) |
| $1 - e^{-4}(1 + 4 + \frac{4^2}{2} + \frac{4^3}{3!})$ | M1 | Expression of correct form (any $\lambda$), no end errors |
| $= 0.567$ or $0.566$ | A1 [3] | OR Use of normal scores M1, standardising M1, standardising with correct cc A1ft, **(ii)** 0.599. Award A mark only if normal given in **(ii)**; OR Binomial M1 expression of correct form allow end error, M1 correct form no end error, A1ft 0.567 or 0.566. Award A mark only if Bin given in **(ii)**. NB Part **(iii)** must be Poisson or ft from **(ii)** for A mark to be awarded. SR If no answer given in **(ii)** allow BOD for A marks. |
---3 A book contains 40000 words. For each word, the probability that it is printed wrongly is 0.0001 and these errors occur independently. The number of words printed wrongly in the book is represented by the random variable $X$.\\
(i) State the exact distribution of $X$, including the values of any parameters.\\
(ii) State an approximate distribution for $X$, including the values of any parameters, and explain why this approximate distribution is appropriate.\\
(iii) Use this approximate distribution to find the probability that there are more than 3 words printed wrongly in the book.
\hfill \mbox{\textit{CAIE S2 2010 Q3 [7]}}