| Exam Board | SPS |
|---|---|
| Module | SPS ASFM Statistics (SPS ASFM Statistics) |
| Year | 2021 |
| Session | May |
| Topic | Poisson distribution |
| Type | Finding maximum n for P(X=0) threshold |
| Difficulty | Moderate -0.3 This is a straightforward Poisson distribution question requiring standard applications: calculating probabilities using the Poisson formula, solving a simple equation involving e^x, and computing expected value. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average but not trivial due to the multi-step nature and need to work with different rates. |
| Spec | 5.02i 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 modelling |
A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.
\begin{enumerate}[label=(\alph*)]
\item Find the probability of exactly 4 faults in a 15 metre length of cloth.
(2)
\item Find the probability of more than 10 faults in 60 metres of cloth.
(3)
\end{enumerate}
A retailer buys a large amount of this cloth and sells it in pieces of length $x$ metres. He chooses $x$ so that the probability of no faults in a piece is 0.80
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Write down an equation for $x$ and show that $x = 1.7$ to 2 significant figures.
(4)
\end{enumerate}
The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of £1.50 on any pieces that do contain faults.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the retailer's expected profit.
(4)
\end{enumerate}
\hfill \mbox{\textit{SPS SPS ASFM Statistics 2021 Q7}}