| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State conditions for Poisson approximation |
| Difficulty | Easy -1.2 Part (a) is pure recall of standard conditions (n large, p small, np moderate). Part (b) is a routine application with given parameters requiring only P(X ≥ 4) = 1 - P(X ≤ 3) using tables or calculator. No problem-solving or insight needed, just standard bookwork and direct calculation. |
| Spec | 2.04d Normal approximation to binomial5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(n\) large | B1 | Accept: \(n\) (number of trials) large/high/big, \(n > 50\), or any number larger than 50 |
| \(p\) small | B1 | Accept: \(p\) (probability) small/close to \(0\), \(p < 0.2\), or any number less than 0.2. Do not accept "low". Must appear in part (a). |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Let \(X\) be the number of letters delivered to the wrong house | — | Definition of variable |
| \(X \sim B(1000, 0.01)\) | — | |
| \(\text{Po}(10)\) | B1 | Award for writing or using \(\text{Po}(10)\) |
| \(P(X \geq 4) = 1 - P(X \leq 3)\) | M1 | Using a Poisson (\(\lambda\) need not equal 10) and writing/using \(1 - P(X \leq 3)\). Do not accept \(1 - P(X < 4)\) unless they have used \(1 - P(X \leq 3)\) |
| \(= 1 - 0.0103\) | — | |
| \(= 0.9897\) | A1 | cao, must be 4 dp |
| (3) | ||
| Total 5 |
## Question 1:
**Part (a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $n$ large | B1 | Accept: $n$ (number of trials) large/high/big, $n > 50$, or any number larger than 50 |
| $p$ small | B1 | Accept: $p$ (probability) small/close to $0$, $p < 0.2$, or any number less than 0.2. Do **not** accept "low". Must appear in part (a). |
| | **(2)** | |
---
**Part (b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Let $X$ be the number of letters delivered to the wrong house | — | Definition of variable |
| $X \sim B(1000, 0.01)$ | — | |
| $\text{Po}(10)$ | B1 | Award for writing or using $\text{Po}(10)$ |
| $P(X \geq 4) = 1 - P(X \leq 3)$ | M1 | Using a Poisson ($\lambda$ need not equal 10) and writing/using $1 - P(X \leq 3)$. Do not accept $1 - P(X < 4)$ unless they have used $1 - P(X \leq 3)$ |
| $= 1 - 0.0103$ | — | |
| $= 0.9897$ | A1 | cao, must be 4 dp |
| | **(3)** | |
| | **Total 5** | |
**NB:**
- An answer of $\approx 0.990$ on its own gains **B0M0A0** unless there is evidence that $\text{Po}(10)$ is used, in which case it gets **B1M1A0**
- Using $B(1000, 0.01)$ gives $0.989927\ldots$ and gains **B0M0A0**\begin{enumerate}
\item (a) Write down the conditions under which the Poisson distribution can be used as an approximation to the binomial distribution.
\end{enumerate}
The probability of any one letter being delivered to the wrong house is 0.01 On a randomly selected day Peter delivers 1000 letters.\\
(b) Using a Poisson approximation, find the probability that Peter delivers at least 4 letters to the wrong house.
Give your answer to 4 decimal places.\\
\hfill \mbox{\textit{Edexcel S2 2013 Q1 [5]}}