| Exam Board | AQA |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Principle of Inclusion/Exclusion |
| Type | Three-Set Venn Diagram Construction |
| Difficulty | Easy -1.2 This is a standard Venn diagram question with three sets requiring routine application of inclusion-exclusion principle to fill regions, followed by basic probability calculations. The most challenging part is the independence test in part (c), but this is a textbook application of P(C∩T) = P(C)×P(T). All steps are algorithmic with no novel insight required, making it easier than average. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables |
| Types of pet | C | D | T | C and D | C and T | D and T | C, D and T |
| Number of households | 153 | 70 | 45 | 48 | 21 | 32 | 17 |
| Answer | Marks | Guidance |
|---|---|---|
| 16(a) | Writes at least two of 4,15, 17 or | |
| 31 in the correct place | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| correct place | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| fully correctly including 56 | 1.1b | A1 |
| Subtotal | 3 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 16(b)(i) | Obtains the correct probability |
| Answer | Marks | Guidance |
|---|---|---|
| fraction or decimals obtained | 1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 16(b)(ii) | Adds their 4 + 15 + 17 + 31 |
| Answer | Marks | Guidance |
|---|---|---|
| stage | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| fraction or decimals obtained | 1.1b | A1 |
| Subtotal | 2 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 16(b)(iii) | Adds their 101 + 31 + 7 |
| Answer | Marks | Guidance |
|---|---|---|
| stage | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| fraction or decimals obtained | 1.1b | A1 |
| Subtotal | 2 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 16(c) | Finds correct P(C) × P(T) |
| Answer | Marks | Guidance |
|---|---|---|
| 𝑃𝑃(𝑇𝑇 | 𝐶𝐶)= 153 | |
| All figures must be correct | 3.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| All figures must be correct | 2.2a | R1 |
| Subtotal | 2 | |
| Question 16 Total | 10 | |
| Q | Marking instructions | AO |
Question 16:
--- 16(a) ---
16(a) | Writes at least two of 4,15, 17 or
31 in the correct place | 1.1a | M1
Obtains either 7 or 9 in the
correct place | 1.1b | A1
Completes the Venn diagram
fully correctly including 56 | 1.1b | A1
Subtotal | 3
Q | Marking instructions | AO | Marks | Typical solution
--- 16(b)(i) ---
16(b)(i) | Obtains the correct probability
AWFW [0.42, 0.421]
Ignore subsequent incorrect
simplification once correct
fraction or decimals obtained | 1.1b | B1 | 1 01
240
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 16(b)(ii) ---
16(b)(ii) | Adds their 4 + 15 + 17 + 31
PI by correct answer OE
The total is not required at this
stage | 1.1a | M1 | 4 + 15 + 17 + 31 = 67
67
240
Obtains the correct probability
AWFW [0.279, 0.28]
Ignore subsequent incorrect
simplification once correct
fraction or decimals obtained | 1.1b | A1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 16(b)(iii) ---
16(b)(iii) | Adds their 101 + 31 + 7
PI by correct answer OE
The total is not required at this
stage | 1.1a | M1 | 101 + 31 + 7 = 139
1 39
195
Obtains the correct probability
AWFW [0.71, 0.713]
Ignore subsequent incorrect
simplification once correct
fraction or decimals obtained | 1.1b | A1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 16(c) ---
16(c) | Finds correct P(C) × P(T)
or
uses correct conditional
probability
eg
21
OE
𝑃𝑃(𝑇𝑇|𝐶𝐶)= 153
All figures must be correct | 3.1b | M1 | 153 45 153
0.12
𝑃𝑃(𝐶𝐶)×𝑃𝑃(𝑇𝑇)= 240×240= 1280
=
21
0.0875
𝑃𝑃(𝐶𝐶∩𝑇𝑇)= 240
0.12 ≠ 0.0=875
Hence events C and T are not
independent
Compares with
21
and concludes owning a cat
𝑃𝑃(𝐶𝐶 ∩𝑇𝑇)= 240
153
and owning a tortoise are not
1280
independent
eg ≠ so not
21 153
independent
240 1280
or
compares conditional probability
eg ≠ so not independent
21 45
153 240
All figures must be correct | 2.2a | R1
Subtotal | 2
Question 16 Total | 10
Q | Marking instructions | AO | Marks | Typical solutionA sample of 240 households were asked which, if any, of the following animals they own as pets:
• cats (C)
• dogs (D)
• tortoises (T)
The results are shown in the table below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Types of pet & C & D & T & C and D & C and T & D and T & C, D and T \\
\hline
Number of households & 153 & 70 & 45 & 48 & 21 & 32 & 17 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Represent this information by fully completing the Venn diagram below.
[3 marks]
\includegraphics{figure_16}
\item A household is chosen at random from the sample.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that the household owns a cat only.
[1 mark]
\item Find the probability that the household owns at least two of the three types of pet.
[2 marks]
\item Find the probability that the household owns a cat or a dog or both, given that the household does not own a tortoise.
[2 marks]
\end{enumerate}
\item Determine whether a household owning a cat and a household owning a tortoise are independent of each other.
Fully justify your answer.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 3 2022 Q16 [10]}}