| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Basic two-way table probability |
| Difficulty | Easy -1.2 This is a straightforward conditional probability question using a two-way table with clearly labeled data. Parts (a)(i)-(iii) require simple division of table values, (a)(iv)-(v) require basic conditional probability (restricting to a subset then calculating), and part (b) applies binomial probability with estimated probabilities from the table. All calculations are routine applications of formulas with no conceptual challenges or novel problem-solving required. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| \multirow{2}{*}{} | Charge band for course of treatment | |||||
| Band 0 | Band 1 | Band 2 | Band 3 | Total | ||
| \multirow{4}{*}{Age of patient (years)} | Under 19 | 32 | 43 | 5 | 0 | 80 |
| Between 19 and 40 | 17 | 62 | 22 | 3 | 104 | |
| Between 41 and 65 | 28 | 82 | 35 | 31 | 176 | |
| 66 or over | 13 | 53 | 68 | 6 | 140 | |
| Total | 90 | 240 | 130 | 40 | 500 | |
3 The table shows, for a random sample of 500 patients attending a dental surgery, the patients' ages, in years, and the NHS charge bands for the patients' courses of treatment. Band 0 denotes the least expensive charge band and band 3 denotes the most expensive charge band.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{4}{|c|}{Charge band for course of treatment} & \\
\hline
& & Band 0 & Band 1 & Band 2 & Band 3 & Total \\
\hline
\multirow{4}{*}{Age of patient (years)} & Under 19 & 32 & 43 & 5 & 0 & 80 \\
\hline
& Between 19 and 40 & 17 & 62 & 22 & 3 & 104 \\
\hline
& Between 41 and 65 & 28 & 82 & 35 & 31 & 176 \\
\hline
& 66 or over & 13 & 53 & 68 & 6 & 140 \\
\hline
& Total & 90 & 240 & 130 & 40 & 500 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate, to three decimal places, the probability that a patient, selected at random from these 500 patients, was:
\begin{enumerate}[label=(\roman*)]
\item aged between 41 and 65;
\item aged 66 or over and charged at band 2;
\item aged between 19 and 40 and charged at most at band 1;
\item aged 41 or over, given that the patient was charged at band 2;
\item charged at least at band 2, given that the patient was not aged 66 or over.
\end{enumerate}\item Four patients at this dental surgery, not included in the above 500 patients, are selected at random.
Estimate, to three significant figures, the probability that two of these four patients are aged between 41 and 65 and are not charged at band 0 , and the other two patients are aged 66 or over and are charged at either band 1 or band 2.\\[0pt]
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2016 Q3 [14]}}