Question 6 - A-Level Maths

AQA S1 2006 June — Question 6

Exam Board	AQA
Module	S1 (Statistics 1)
Year	2006
Session	June
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Probability Definitions
Type	Two-way table probabilities
Difficulty	Easy -1.3 This is a straightforward two-way table question requiring only direct reading from the table and application of basic probability definitions (intersection, union, conditional probability, mutual exclusivity, independence). All calculations are simple fractions with no complex reasoning or problem-solving required—purely routine recall and application of AS-level probability formulas.
Spec	2.03a Mutually exclusive and independent events 2.03b Probability diagrams: tree, Venn, sample space 2.03c Conditional probability: using diagrams/tables 2.03d Calculate conditional probability: from first principles

6 A housing estate consists of 320 houses: 120 detached and 200 semi-detached. The numbers of children living in these houses are shown in the table.

\multirow{2}{*}{}	Number of children
	None	One	Two	At least three	Total
Detached house	24	32	41	23	120
Semi-detached house	40	37	88	35	200
Total	64	69	129	58	320

A house on the estate is selected at random. \(D\) denotes the event 'the house is detached'. \(R\) denotes the event 'no children live in the house'. \(S\) denotes the event 'one child lives in the house'. \(T\) denotes the event 'two children live in the house'.
( \(D ^ { \prime }\) denotes the event 'not \(D\) '.)

Find:
1. \(\mathrm { P } ( D )\);
2. \(\quad \mathrm { P } ( D \cap R )\);
3. \(\quad \mathrm { P } ( D \cup T )\);
4. \(\mathrm { P } ( D \mid R )\);
5. \(\mathrm { P } \left( R \mid D ^ { \prime } \right)\).
1. Name two of the events \(D , R , S\) and \(T\) that are mutually exclusive.
2. Determine whether the events \(D\) and \(R\) are independent. Justify your answer.
Define, in the context of this question, the event:
1. \(D ^ { \prime } \cup T\);
2. \(D \cap ( R \cup S )\).

Show mark scheme Show mark scheme source

Answer	Marks	Guidance
\(O(R)\)	\(1(S)\)	\(2(T)\)
\(D(D)\)	24	32
\(D(D')\)	40	37
\(T\)	64	69

Question 6(a)(i)

Answer	Marks	Guidance
P(\(D\)) = \(\frac{120}{320}\) or \(\frac{3}{8}\) or \(0.375\)	B1	1 mark; CAO; or equivalent

Total: 1 mark

Question 6(a)(ii)

Answer	Marks	Guidance
P(\(D \cap R\)) = \(\frac{24}{320}\) or \(\frac{3}{40}\) or \(0.075\)	B1	1 mark; CSO; or equivalent

Total: 1 mark

Question 6(a)(iii)

Answer	Marks	Guidance
P(\(D \cup T\)) = \(\frac{120 + 88}{320} = \frac{129 + 24 + 32 + 23}{320} = \frac{208}{320}\) or \(\frac{13}{20}\) or \(0.65\)	M1, A1	2 marks; CAO; or equivalent

Total: 2 marks

Question 6(a)(iv)

Answer	Marks	Guidance
P(\(D	R\)) = \(\frac{P(D \cap R)}{P(R)} = \frac{\text{(ii)}}{\text{P}(R)} = \frac{24/(320)}{64/(320)} = \frac{24}{64}\) or \(\frac{3}{8}\) or \(0.375\)	M1, A1

Total: 2 marks

Question 6(a)(v)

Answer	Marks	Guidance
P(\(R	D'\)) = \(\frac{P(R \cap D')}{P(D')} = \frac{40/(320)}{200/(320)} = \frac{40}{200}\) or \(\frac{1}{5}\) or \(0.2\)	M1, M1, A1

Total: 3 marks

Question 6(b)(i)

Answer	Marks	Guidance
\(R\) and \(S\) or \(R\) and \(T\) or \(S\) and \(T\)	B1	1 mark; not \(D\) and \(D'\)

Total: 1 mark

Question 6(b)(ii)

Answer	Marks	Guidance
P(\(D\)) = \(0.375\) = P(\(D	R\)) or (i) = (iv) so YES	M1, A1

Total: 2 marks

Question 6(c)(i)

Answer	Marks	Guidance
A semi-detached house or two children (or both)	B1, B1	2 marks; CAO; or equivalent

Total: 2 marks

Question 6(c)(ii)

Answer	Marks	Guidance
A detached house and/with less than two children	B1, B1	2 marks; CAO; (0 or 1 must not include 'both')

Total: 2 marks

TOTAL: 75 marks

| | $O(R)$ | $1(S)$ | $2(T)$ | $\geq 3$ | T |
|---|---|---|---|---|---|
| $D(D)$ | 24 | 32 | 41 | 23 | 120 |
| $D(D')$ | 40 | 37 | 88 | 35 | 200 |
| $T$ | 64 | 69 | 129 | 58 | 320 |

## Question 6(a)(i)

P($D$) = $\frac{120}{320}$ or $\frac{3}{8}$ or $0.375$ | B1 | 1 mark; CAO; or equivalent |

**Total: 1 mark**

## Question 6(a)(ii)

P($D \cap R$) = $\frac{24}{320}$ or $\frac{3}{40}$ or $0.075$ | B1 | 1 mark; CSO; or equivalent |

**Total: 1 mark**

## Question 6(a)(iii)

P($D \cup T$) = $\frac{120 + 88}{320} = \frac{129 + 24 + 32 + 23}{320} = \frac{208}{320}$ or $\frac{13}{20}$ or $0.65$ | M1, A1 | 2 marks; CAO; or equivalent |

**Total: 2 marks**

## Question 6(a)(iv)

P($D | R$) = $\frac{P(D \cap R)}{P(R)} = \frac{\text{(ii)}}{\text{P}(R)} = \frac{24/(320)}{64/(320)} = \frac{24}{64}$ or $\frac{3}{8}$ or $0.375$ | M1, A1 | 2 marks; CAO; or equivalent; M0 if independence assumed |

**Total: 2 marks**

## Question 6(a)(v)

P($R | D'$) = $\frac{P(R \cap D')}{P(D')} = \frac{40/(320)}{200/(320)} = \frac{40}{200}$ or $\frac{1}{5}$ or $0.2$ | M1, M1, A1 | 3 marks; CAO; or equivalent; allow independence assumed numerator; denominator |

**Total: 3 marks**

## Question 6(b)(i)

$R$ and $S$ or $R$ and $T$ or $S$ and $T$ | B1 | 1 mark; not $D$ and $D'$ |

**Total: 1 mark**

## Question 6(b)(ii)

P($D$) = $0.375$ = P($D | R$) or (i) = (iv) so YES | M1, A1 | 2 marks; P($D$) $\times$ P($R$) = $0.375 \times 0.2 = 0.075$ = P($D \cap R$) or (ii); or P($R | D$) = P($R$) = 0.2, etc |

**Total: 2 marks**

## Question 6(c)(i)

A semi-detached house or two children (or both) | B1, B1 | 2 marks; CAO; or equivalent |

**Total: 2 marks**

## Question 6(c)(ii)

A detached house and/with less than two children | B1, B1 | 2 marks; CAO; (0 or 1 must not include 'both') |

**Total: 2 marks**

---

**TOTAL: 75 marks**

Show LaTeX source

6 A housing estate consists of 320 houses: 120 detached and 200 semi-detached. The numbers of children living in these houses are shown in the table.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multirow{2}{*}{} & \multicolumn{4}{|c|}{Number of children} &  \\
\hline
 & None & One & Two & At least three & Total \\
\hline
Detached house & 24 & 32 & 41 & 23 & 120 \\
\hline
Semi-detached house & 40 & 37 & 88 & 35 & 200 \\
\hline
Total & 64 & 69 & 129 & 58 & 320 \\
\hline
\end{tabular}
\end{center}

A house on the estate is selected at random.\\
$D$ denotes the event 'the house is detached'.\\
$R$ denotes the event 'no children live in the house'.\\
$S$ denotes the event 'one child lives in the house'.\\
$T$ denotes the event 'two children live in the house'.\\
( $D ^ { \prime }$ denotes the event 'not $D$ '.)
\begin{enumerate}[label=(\alph*)]
\item Find:
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { P } ( D )$;
\item $\quad \mathrm { P } ( D \cap R )$;
\item $\quad \mathrm { P } ( D \cup T )$;
\item $\mathrm { P } ( D \mid R )$;
\item $\mathrm { P } \left( R \mid D ^ { \prime } \right)$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Name two of the events $D , R , S$ and $T$ that are mutually exclusive.
\item Determine whether the events $D$ and $R$ are independent. Justify your answer.
\end{enumerate}\item Define, in the context of this question, the event:
\begin{enumerate}[label=(\roman*)]
\item $D ^ { \prime } \cup T$;
\item $D \cap ( R \cup S )$.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S1 2006 Q6}}

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