| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Combined independent Poisson probabilities |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson approximation to binomial with standard probability calculations. Part (a) requires recognizing that defective spoons and forks follow independent Poisson distributions, then computing basic probabilities using complement rule and addition. Part (b) involves simple algebraic manipulation of Poisson probability formulas. All techniques are routine for S2 level with no novel insight required, making it slightly easier than average. |
| Spec | 5.02d Binomial: mean np and variance np(1-p)5.02i 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/Working | Mark | Guidance |
| \(0.01 \times 80\) and \(0.015 \times 60\) | M1 | \((1 - e^{-\lambda}) \times (1 - e^{-\mu})\) any \(\lambda\), \(\mu\) \((\lambda \neq \mu)\) |
| \((1 - e^{-0.8}) \times (1 - e^{-0.9})\) | M1 | allow one end error |
| \(= 0.327\) (3 sf) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\lambda = 0.02 \times 40 + 0.015 \times 60\) | M1 | or their \(0.8 + 0.9\) |
| \(e^{-1.7} \times \left(1 + 1.7 + \frac{1.7^2}{2}\right)\) | M1 | |
| \(= 0.757\) (3 sf) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(e^{-\lambda} \times \lambda = p\) and \(e^{-\lambda} \times \frac{\lambda^2}{2} = 1.5p\) | M1 | or \(e^{-\lambda} \times \frac{\lambda^2}{2} = 1.5 \times e^{-\lambda} \times \lambda\) seen or implied |
| \(\lambda = 3\) | A1 | |
| \(p = e^{-3} \times 3\) | M1 | their \(\lambda\) |
| \(= 0.149\) (3 sf) | A1 [4] |
## Question 7:
### Part (a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.01 \times 80$ and $0.015 \times 60$ | M1 | $(1 - e^{-\lambda}) \times (1 - e^{-\mu})$ any $\lambda$, $\mu$ $(\lambda \neq \mu)$ |
| $(1 - e^{-0.8}) \times (1 - e^{-0.9})$ | M1 | allow one end error |
| $= 0.327$ (3 sf) | A1 [3] | |
### Part (a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\lambda = 0.02 \times 40 + 0.015 \times 60$ | M1 | or their $0.8 + 0.9$ |
| $e^{-1.7} \times \left(1 + 1.7 + \frac{1.7^2}{2}\right)$ | M1 | |
| $= 0.757$ (3 sf) | A1 [3] | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $e^{-\lambda} \times \lambda = p$ and $e^{-\lambda} \times \frac{\lambda^2}{2} = 1.5p$ | M1 | or $e^{-\lambda} \times \frac{\lambda^2}{2} = 1.5 \times e^{-\lambda} \times \lambda$ seen or implied |
| $\lambda = 3$ | A1 | |
| $p = e^{-3} \times 3$ | M1 | their $\lambda$ |
| $= 0.149$ (3 sf) | A1 [4] | |7
\begin{enumerate}[label=(\alph*)]
\item A large number of spoons and forks made in a factory are inspected. It is found that $1 \%$ of the spoons and $1.5 \%$ of the forks are defective. A random sample of 140 items, consisting of 80 spoons and 60 forks, is chosen. Use the Poisson approximation to the binomial distribution to find the probability that the sample contains
\begin{enumerate}[label=(\roman*)]
\item at least 1 defective spoon and at least 1 defective fork,
\item fewer than 3 defective items.
\end{enumerate}\item The random variable $X$ has the distribution $\operatorname { Po } ( \lambda )$. It is given that
$$\mathrm { P } ( X = 1 ) = p \quad \text { and } \quad \mathrm { P } ( X = 2 ) = 1.5 p$$
where $p$ is a non-zero constant. Find the value of $\lambda$ and hence find the value of $p$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2016 Q7 [10]}}