| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Piecewise or conditional probability function |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring basic probability distribution properties (sum to 1), simple probability calculations, and standard application of E(aX+b) and SD(aX+b) formulas. All steps are routine with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
6 A small supermarket has a total of four checkouts, at least one of which is always staffed. The probability distribution for $R$, the number of checkouts that are staffed at any given time, is
$$\mathrm { P } ( R = r ) = \left\{ \begin{array} { c l }
\frac { 2 } { 3 } \left( \frac { 1 } { 3 } \right) ^ { r - 1 } & r = 1,2,3 \\
k & r = 4
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac { 1 } { 27 }$.
\item Find the probability that, at any given time, there will be at least 3 checkouts that are staffed.
\item It is suggested that the total number of customers, $C$, that can be served at the checkouts per hour may be modelled by
$$C = 27 R + 5$$
Find:
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { E } ( C )$;
\item the standard deviation of $C$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2009 Q6 [10]}}