| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with normal approximation. Part (a) requires calculating a Poisson probability with λ=10, which is routine. Part (b) involves recognizing when to use normal approximation (large λ=40) and applying continuity correction—standard S2 material requiring minimal problem-solving beyond following the established procedure. |
| Spec | 2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(X \sim \text{Po}(10)\) | B1 | \(P(X < 9) = P(X \leq 8) = 0.3328\) |
| (b) \(Y \sim \text{Po}(40)\) | B1 | \(Y\) is approximately \(N(40, 40)\) |
| Total [9] |
**(a)** $X \sim \text{Po}(10)$ | B1 | $P(X < 9) = P(X \leq 8) = 0.3328$ | M1 A1 | (3 marks)
**(b)** $Y \sim \text{Po}(40)$ | B1 | $Y$ is approximately $N(40, 40)$ | M1 A1 | $P(Y > 50) = 1 - P(Y \leq 50) = 1 - P\left(Z < \frac{50.5 - 40}{\sqrt{40}}\right) = 1 - P(Z < 1.660\ldots) = 1 - 0.9515 = 0.0485$ | M1 M1 A1 | (6 marks)
| **Total [9]**
---\begin{enumerate}
\item A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.
\end{enumerate}
Find the probability that\\
(a) fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am.
The café serves breakfast every day between 8 am and 12 noon.\\
(b) Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.\\
\hfill \mbox{\textit{Edexcel S2 2010 Q5 [9]}}