| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Principle of Inclusion/Exclusion |
| Type | Two-Way Table to Probability |
| Difficulty | Moderate -0.8 This is a straightforward S1 question testing basic probability from two-way tables: reading values, calculating simple and conditional probabilities, checking independence, and a standard multinomial probability calculation. All parts require only direct application of formulas with no problem-solving insight or novel reasoning—easier than average A-level maths. |
| Spec | 2.03a Mutually exclusive and independent events2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles2.04c Calculate binomial probabilities |
| \multirow{2}{*}{} | Number of toilets | |||||
| 1 | 2 | 3 | 4 or more | Total | ||
| \multirow{5}{*}{Number of bedrooms} | 1 | 46 | 14 | 0 | 0 | 60 |
| 2 | 24 | 67 | 23 | 0 | 114 | |
| 3 | 7 | 72 | 99 | 16 | 194 | |
| 4 | 0 | 19 | 123 | 48 | 190 | |
| 5 or more | 0 | 0 | 11 | 71 | 82 | |
| Total | 77 | 172 | 256 | 135 | 640 | |
4 A survey of the 640 properties on an estate was undertaken. Part of the information collected related to the number of bedrooms and the number of toilets in each property.
This information is shown in the table.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{4}{|c|}{Number of toilets} & \\
\hline
& & 1 & 2 & 3 & 4 or more & Total \\
\hline
\multirow{5}{*}{Number of bedrooms} & 1 & 46 & 14 & 0 & 0 & 60 \\
\hline
& 2 & 24 & 67 & 23 & 0 & 114 \\
\hline
& 3 & 7 & 72 & 99 & 16 & 194 \\
\hline
& 4 & 0 & 19 & 123 & 48 & 190 \\
\hline
& 5 or more & 0 & 0 & 11 & 71 & 82 \\
\hline
& Total & 77 & 172 & 256 & 135 & 640 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item A property on the estate is selected at random.
Find, giving your answer to three decimal places, the probability that the property has:
\begin{enumerate}[label=(\roman*)]
\item exactly 3 bedrooms;
\item at least 2 toilets;
\item exactly 3 bedrooms and at least 2 toilets;
\item at most 3 bedrooms, given that it has exactly 2 toilets.
\end{enumerate}\item Use relevant answers from part (a) to show that the number of toilets is not independent of the number of bedrooms.
\item Three properties are selected at random from those on the estate which have exactly 3 bedrooms.
Calculate the probability that one property has 2 toilets, one has 3 toilets and the other has at least 4 toilets. Give your answer to three decimal places.
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2012 Q4 [14]}}