| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson with binomial combination |
| Difficulty | Moderate -0.3 This is a straightforward application of Poisson distribution (parts a-b) followed by binomial distribution (part c). All parts require direct formula substitution with no conceptual challenges—students need only identify the correct distributions and apply standard calculations. Slightly easier than average due to the routine nature of the problem. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02i 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 modelling5.02m Poisson: mean = variance = lambda |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | let \(X =\) no. of complaints per day \(\therefore X \sim \text{Po}(6)\) | M1 |
| \(P(X = 3) = 0.1512 - 0.0620 = 0.0892\) | M1 A1 | |
| (b) | \(P(X \geq 10) = 1 - P(X \leq 9) = 1 - 0.9161 = 0.0839\) | M1 A1 |
| (c) | let \(Y =\) no. of days with 10 or more complaints \(\therefore Y \sim B(6, 0.0839)\) | M1 |
| \(P(Y \leq 1) = (0.9161)^6 + 6(0.0839)(0.9161)^5\) | M1 A1 | |
| \(= 0.916 \text{ (3sf)}\) | A1 | (9) |
(a) | let $X =$ no. of complaints per day $\therefore X \sim \text{Po}(6)$ | M1 |
| $P(X = 3) = 0.1512 - 0.0620 = 0.0892$ | M1 A1 |
(b) | $P(X \geq 10) = 1 - P(X \leq 9) = 1 - 0.9161 = 0.0839$ | M1 A1 |
(c) | let $Y =$ no. of days with 10 or more complaints $\therefore Y \sim B(6, 0.0839)$ | M1 |
| $P(Y \leq 1) = (0.9161)^6 + 6(0.0839)(0.9161)^5$ | M1 A1 |
| $= 0.916 \text{ (3sf)}$ | A1 | (9)The manager of a supermarket receives an average of 6 complaints per day from customers.
Find the probability that on one day she receives
\begin{enumerate}[label=(\alph*)]
\item 3 complaints, [3 marks]
\item 10 or more complaints. [2 marks]
\end{enumerate}
The supermarket is open on six days each week.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the probability that the manager receives 10 or more complaints on no more than one day in a week. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q2 [9]}}