| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | November |
| 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 -0.3 This is a straightforward application of the Poisson approximation to the binomial distribution with standard conditions (large n, small p). Part (a) requires a single Poisson probability calculation, part (b) tests recall of the justification criteria (n large, p small, np moderate), and part (c) extends to summing two Poisson distributions. All steps are routine textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Po}\left(\frac{2}{3}\right)\) | B1 | Poisson with correct mean stated (to at least 3 sf) or implied in working |
| \(1 - e^{-\frac{2}{3}}\left(1 + \frac{2}{3}\right)\) | M1 | \(1 - P(X = 0 \text{ or } 1)\); allow incorrect \(\lambda\); allow one end error |
| \(= 0.144\) (3 sf) | A1 | SC B1 for use of binomial or no working shown leading to correct final answer |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(n > 50\) and \(np = \frac{2}{3} < 5\) or \(n > 50\) and \(p = \frac{1}{300} < 0.1\) | B1 | Accept \(p\) or \(np\) clearly stated in part (a). Do not accept \(n\) is large and \(p\) is small |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Po}\left(\frac{11}{3}\right)\) | B1 | Poisson with correct mean stated (to at least 3 sf) or implied in working |
| \(e^{-\frac{11}{3}}\left(1 + \frac{11}{3} + \frac{\left(\frac{11}{3}\right)^2}{2!} + \frac{\left(\frac{11}{3}\right)^3}{3!}\right)\) | M1 | \(P(X = 0, 1, 2, 3)\); allow incorrect \(\lambda\); allow one end error. Must not be multiplied by any additional values |
| \(= 0.501\) (3 sf) | A1 | As final answer |
| 3 |
## Question 1:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Po}\left(\frac{2}{3}\right)$ | **B1** | Poisson with correct mean stated (to at least 3 sf) or implied in working |
| $1 - e^{-\frac{2}{3}}\left(1 + \frac{2}{3}\right)$ | **M1** | $1 - P(X = 0 \text{ or } 1)$; allow incorrect $\lambda$; allow one end error |
| $= 0.144$ (3 sf) | **A1** | SC B1 for use of binomial or no working shown leading to correct final answer |
| | **3** | |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $n > 50$ and $np = \frac{2}{3} < 5$ **or** $n > 50$ and $p = \frac{1}{300} < 0.1$ | **B1** | Accept $p$ or $np$ clearly stated in part **(a)**. Do not accept $n$ is large and $p$ is small |
| | **3** | |
### Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Po}\left(\frac{11}{3}\right)$ | **B1** | Poisson with correct mean stated (to at least 3 sf) or implied in working |
| $e^{-\frac{11}{3}}\left(1 + \frac{11}{3} + \frac{\left(\frac{11}{3}\right)^2}{2!} + \frac{\left(\frac{11}{3}\right)^3}{3!}\right)$ | **M1** | $P(X = 0, 1, 2, 3)$; allow incorrect $\lambda$; allow one end error. Must not be multiplied by any additional values |
| $= 0.501$ (3 sf) | **A1** | As final answer |
| | **3** | |1 It is known that, on average, 1 in 300 flowers of a certain kind are white. A random sample of 200 flowers of this kind is selected.
\begin{enumerate}[label=(\alph*)]
\item Use an appropriate approximating distribution to find the probability that more than 1 flower in the sample is white.
\item Justify the approximating distribution used in part (a).\\
The probability that a randomly chosen flower of another kind is white is 0.02 . A random sample of 150 of these flowers is selected.
\item Use an appropriate approximating distribution to find the probability that the total number of white flowers in the two samples is less than 4 .
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q1 [7]}}