| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | March |
| 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 -0.3 This is a straightforward application of the Poisson approximation to the binomial distribution. Parts (a)-(c) involve identifying B(1000, 1/400), approximating with Po(2.5), and calculating standard Poisson probabilities from tables. Part (d) requires recognizing Po(1.75) and using P(X≥1)=1-P(X=0). All steps are routine with no problem-solving insight required, making it slightly easier than average. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(B\!\left(1000, \frac{1}{400}\right)\) | B1 | Accept Bin and \(n=1000\), \(p=\frac{1}{400}\) |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(Po(2.5)\) | B2 | B1 for Po. B1 for \(\lambda=2.5\) |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(e^{-2.5}\times\frac{2.5^4}{4!}\) | M1 | FT *their* (b) for Normal must have a continuity correction. Allow any \(\lambda\) |
| \(0.134\) (3 sf) | A1 | CWO |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(e^{-2.5}\!\left(\frac{2.5^2}{2!}+\frac{2.5^3}{3!}+\frac{2.5^4}{4!}\right)\) | M1 | FT *their* (b) for Normal must have a continuity correction. Allow with one term extra or omitted or wrong. Allow any \(\lambda\) |
| \(0.604\) (3 sf) | A1 | CWO |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\lambda = 2.5\times 0.7\) or \(\lambda = 700\times\frac{1}{400}\ [=1.75]\) | M1 | Must see \(\lambda\) or use of Poisson |
| \(1 - e^{-1.75}\) | M1 | Allow any \(\lambda\). Allow \(1-P(0,1)\) |
| \(0.826\) | A1 | SC B1 Use of \(B(700, 0.0025)\) leading to \(0.826\) |
| 3 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $B\!\left(1000, \frac{1}{400}\right)$ | B1 | Accept Bin and $n=1000$, $p=\frac{1}{400}$ |
| | **1** | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $Po(2.5)$ | B2 | B1 for Po. B1 for $\lambda=2.5$ |
| | **2** | |
## Question 4(c)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $e^{-2.5}\times\frac{2.5^4}{4!}$ | M1 | FT *their* **(b)** for Normal must have a continuity correction. Allow any $\lambda$ |
| $0.134$ (3 sf) | A1 | CWO |
| | **2** | |
## Question 4(c)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $e^{-2.5}\!\left(\frac{2.5^2}{2!}+\frac{2.5^3}{3!}+\frac{2.5^4}{4!}\right)$ | M1 | FT *their* **(b)** for Normal must have a continuity correction. Allow with one term extra or omitted or wrong. Allow any $\lambda$ |
| $0.604$ (3 sf) | A1 | CWO |
| | **2** | |
## Question 4(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\lambda = 2.5\times 0.7$ or $\lambda = 700\times\frac{1}{400}\ [=1.75]$ | M1 | Must see $\lambda$ or use of Poisson |
| $1 - e^{-1.75}$ | M1 | Allow any $\lambda$. Allow $1-P(0,1)$ |
| $0.826$ | A1 | **SC B1** Use of $B(700, 0.0025)$ leading to $0.826$ |
| | **3** | |4 On average, 1 in 400 microchips made at a certain factory are faulty. The number of faulty microchips in a random sample of 1000 is denoted by $X$.
\begin{enumerate}[label=(\alph*)]
\item State the distribution of $X$, giving the values of any parameters.
\item State an approximating distribution for $X$, giving the values of any parameters.
\item Use this approximating distribution to find each of the following.
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { P } ( X = 4 )$.
\item $\mathrm { P } ( 2 \leqslant X \leqslant 4 )$.
\end{enumerate}\item Use a suitable approximating distribution to find the probability that, in a random sample of 700 microchips, there will be at least 1 faulty one.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q4 [10]}}