| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| 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 straightforward application of the Poisson approximation to the binomial distribution. Students need to recognize the conditions (n large, p small, np moderate), calculate λ = np = 4, then apply standard Poisson probability formulas. The justification and calculation are routine textbook exercises with no novel problem-solving required. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| \(X \sim B(200, 0.02)\) | B1 | Implied conditions |
| \(n\) large, \(P\) small so \(X \sim \text{Po}(np) = \text{Po}(4)\) | B1, B1 | \(P_0(4)\) |
| \(P(X=5) = \frac{e^{-4}4^5}{5!} = 0.1563\) | M1 | \(P(X \le 5) - P(X \le 4)\) |
| A1 | 0.1563 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X < 5) = P(X \le 4) = 0.6288\) | M1 | \(P(X \le 4)\) |
| A1 | 0.6288 |
**5(a)**
| $X \sim B(200, 0.02)$ | B1 | Implied conditions |
| $n$ large, $P$ small so $X \sim \text{Po}(np) = \text{Po}(4)$ | B1, B1 | $P_0(4)$ |
| $P(X=5) = \frac{e^{-4}4^5}{5!} = 0.1563$ | M1 | $P(X \le 5) - P(X \le 4)$ |
| | A1 | 0.1563 |
**5(b)**
| $P(X < 5) = P(X \le 4) = 0.6288$ | M1 | $P(X \le 4)$ |
| | A1 | 0.6288 |
---5. In a manufacturing process, $2 \%$ of the articles produced are defective. A batch of 200 articles is selected.
\begin{enumerate}[label=(\alph*)]
\item Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.
\item Estimate the probability there are less than 5 defective articles.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2005 Q5 [7]}}