| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2004 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Calculate multiple probabilities using Poisson approximation |
| Difficulty | Moderate -0.8 This is a straightforward textbook question on Poisson approximation to binomial. Part (a) requires simple recall of conditions (n large, p small). Parts (b) and (c) are routine calculations: (b) uses binomial directly with small n, (c) applies Poisson approximation with λ=np=8 and requires cumulative probability lookup. No problem-solving insight needed, just standard procedure application. |
| Spec | 2.04c Calculate binomial probabilities2.04d Normal approximation to binomial5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(n\) large, \(p\) small | B1, B1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(X\) represent the number of people catching the virus, \(X \sim B\left(12, \frac{1}{150}\right)\) | Implied B1 | |
| \(P(X = 2) = C_2^{12}\left(\frac{1}{150}\right)^2\left(\frac{149}{150}\right)^{10} = 0.0027\) | M1A1, A1 | Use of Bin including \(C_2^{12}\), 0.0027(4) only |
| Answer | Marks |
|---|---|
| \(X \sim \text{Po}(np) = \text{Po}(8)\) | Poisson, 8 B1, B1 |
| \(P(X < 7) = P(X \leq 6) = 0.3134\) | \(X \leq 6\) for method, 0.3134 M1A1 |
## Part (a)
$n$ large, $p$ small | B1, B1 | (2 marks)
## Part (b)
Let $X$ represent the number of people catching the virus, $X \sim B\left(12, \frac{1}{150}\right)$ | Implied B1 |
$P(X = 2) = C_2^{12}\left(\frac{1}{150}\right)^2\left(\frac{149}{150}\right)^{10} = 0.0027$ | M1A1, A1 | Use of Bin including $C_2^{12}$, 0.0027(4) only
(4 marks)
## Part (c)
$X \sim \text{Po}(np) = \text{Po}(8)$ | Poisson, 8 B1, B1 |
$P(X < 7) = P(X \leq 6) = 0.3134$ | $X \leq 6$ for method, 0.3134 M1A1 |
(4 marks)
**Total 10 Marks**
---\begin{enumerate}[label=(\alph*)]
\item Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. [2]
\end{enumerate}
A researcher has suggested that 1 in 150 people is likely to catch a particular virus.
Assuming that a person catching the virus is independent of any other person catching it,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the probability that in a random sample of 12 people, exactly 2 of them catch the virus. [4]
\item Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2004 Q4 [10]}}