| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | Normal approximation to sum |
| Difficulty | Standard +0.3 This is a straightforward S2 Poisson distribution question with standard parts: stating conditions (bookwork), direct Poisson probability calculations with given λ, scaling λ for different volumes, and a routine normal approximation for large λ. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
In a certain fluid, bacteria are distributed randomly and occur at a constant average rate of 2.5 in every 10 ml of the fluid.
\begin{enumerate}[label=(\roman*)]
\item State a further condition needed for the number of bacteria in a fixed volume of the fluid to be well modelled by a Poisson distribution, explaining what your answer means. [2]
\end{enumerate}
Assume now that a Poisson model is appropriate.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the probability that in 10 ml there are at least 5 bacteria. [2]
\item Find the probability that in 3.7 ml there are exactly 2 bacteria. [3]
\item Use a suitable approximation to find the probability that in 1000 ml there are fewer than 240 bacteria, justifying your approximation. [7]
\end{enumerate}
\hfill \mbox{\textit{OCR S2 2012 Q8 [14]}}