| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Calculate multiple probabilities using Poisson approximation |
| Difficulty | Moderate -0.8 This is a standard S2 textbook question on binomial-Poisson approximation. Parts (a) and (c) test recall of conditions, while parts (b) and (d) involve routine calculations with given parameters (n=120, p=0.008). The comparison in part (d) is straightforward. No problem-solving or novel insight required—purely procedural application of well-rehearsed techniques. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | fixed no. of eggs, eggs either broken or not, prob. of each egg being broken is same (assuming no accident breaking group together) | B3 |
| (b) | let \(X =\) no. of eggs broken in delivery \(\therefore X \sim B(120, 0.008)\) | M1 |
| \(P(X \leq 1) = (0.992)^{120} + 120(0.008)(0.992)^{119}\) | M1 A1 | |
| \(= 0.7505 \text{ (4sf)}\) | A1 | |
| (c) | \(n\) large, \(p\) small | B1 |
| (d) | \(X \approx \text{Po}(0.96)\) | M1 |
| \(P(X \leq 1) = e^{-0.96}(1 + 0.96)\) | M1 A1 | |
| \(= 0.7505 \text{ (4sf)}\) | A1 | |
| same value to 4sf, very good approx. for these parameters | B1 | (13) |
(a) | fixed no. of eggs, eggs either broken or not, prob. of each egg being broken is same (assuming no accident breaking group together) | B3 |
(b) | let $X =$ no. of eggs broken in delivery $\therefore X \sim B(120, 0.008)$ | M1 |
| $P(X \leq 1) = (0.992)^{120} + 120(0.008)(0.992)^{119}$ | M1 A1 |
| $= 0.7505 \text{ (4sf)}$ | A1 |
(c) | $n$ large, $p$ small | B1 |
(d) | $X \approx \text{Po}(0.96)$ | M1 |
| $P(X \leq 1) = e^{-0.96}(1 + 0.96)$ | M1 A1 |
| $= 0.7505 \text{ (4sf)}$ | A1 |
| same value to 4sf, very good approx. for these parameters | B1 | (13)A shop receives weekly deliveries of 120 eggs from a local farm. The proportion of eggs received from the farm that are broken is 0.008
\begin{enumerate}[label=(\alph*)]
\item Explain why it is reasonable to use the binomial distribution to model the number of eggs that are broken in each delivery. [3 marks]
\item Use the binomial distribution to calculate the probability that at most one egg in a delivery will be broken. [4 marks]
\item State the conditions under which the binomial distribution can be approximated by the Poisson distribution. [1 mark]
\item Using the Poisson approximation to the binomial, find the probability that at most one egg in a delivery will be broken. Comment on your answer. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q6 [13]}}