| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Calculate multiple probabilities using Poisson approximation |
| Difficulty | Moderate -0.3 This is a straightforward application of binomial distribution with clear parameters (n=10, p=0.01) for parts (a) and (b), followed by a standard Poisson approximation (λ=2.5) for part (c). The question explicitly signals the approximation is needed and all calculations are routine with no conceptual challenges beyond recognizing when to use each distribution. |
| Spec | 2.04c Calculate binomial probabilities5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X = 1) = (0.99)^9(0.01) \times 10 = 0.0914\) | M1A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \geq 2) = 1 - P(X \leq 1) = 1 - (p)^{10} - (a) = 0.0043\) | M1, A1/, A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim \text{Po}(2.5)\) | B1B1 | |
| \(P(1 \leq X \leq 4) = P(X \leq 4) - P(X = 0) = 0.8912 - 0.0821 = 0.809\) | M1, A1 | (4 marks) |
## Part (a)
$X$ represents the number of defective components.
$P(X = 1) = (0.99)^9(0.01) \times 10 = 0.0914$ | M1A1 | (2 marks)
## Part (b)
$P(X \geq 2) = 1 - P(X \leq 1) = 1 - (p)^{10} - (a) = 0.0043$ | M1, A1/, A1 | (3 marks)
## Part (c)
$X \sim \text{Po}(2.5)$ | B1B1 |
$P(1 \leq X \leq 4) = P(X \leq 4) - P(X = 0) = 0.8912 - 0.0821 = 0.809$ | M1, A1 | (4 marks)
**Note:** Normal distribution used. B1for mean only
**Special case for parts a and b:**
If they use 0.1 do not treat as misread as it makes it easier.
- (a) M1 A0 if they have 0.3874
- (b) M1 A1ft A0 they will get 0.2639
- (c) Could get B1 B0 M1 A0
**For any other values of $p$ which are in the table:**
Do not use misread. Check using the tables. They could get (a) M1 A0 (b) M1 A1ft A0 (c) B1 B0 M1 A0
---A factory produces components of which 1\% are defective. The components are packed in boxes of 10. A box is selected at random.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the box contains exactly one defective component. [2]
\item Find the probability that there are at least 2 defective components in the box. [3]
\item Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2009 Q5 [9]}}