June 2022 Paper 3 Q16

AQACurrent spec10 marksProbability

16 A 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.

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

(a) Represent this information by fully completing the Venn diagram below. [3 marks]

Venn diagram for completion: three overlapping circles C, D and T inside a rectangle; 101 is already written in the region for C only

(b) A household is chosen at random from the sample.

(i) Find the probability that the household owns a cat only. [1 mark]

(ii) Find the probability that the household owns at least two of the three types of pet. [2 marks]

(iii) 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]

Fully justify your answer. [2 marks]

Completed Venn diagram: C only 101, C and D only 31, D only 7, C, D and T 17, C and T only 4, D and T only 15, T only 9, outside all circles 56 — Mark scheme (a)

Scheme	Marks	AO
Writes at least two of 4, 15, 17 or 31 in the correct place	M1	1.1a
Obtains either 7 or 9 in the correct place	A1	1.1b
Completes the Venn diagram fully correctly including 56	A1	1.1b
	(3)

Small Venn diagram with the regions in two or three of the circles shaded — Mark scheme (b)

Small Venn diagram with the parts of C and D outside T shaded — Mark scheme (b)

Scheme	Marks	AO
(i) Obtains the correct probability AWFW [0.42, 0.421] Ignore subsequent incorrect simplification once correct fraction or decimals obtained	B1	1.1b
	(1)
(ii) Adds their 4 + 15 + 17 + 31 PI by correct answer OE The total is not required at this stage	M1	1.1a
Obtains the correct probability AWFW [0.279, 0.28] Ignore subsequent incorrect simplification once correct fraction or decimals obtained	A1	1.1b
	(2)
(iii) Adds their 101 + 31 + 7 PI by correct answer OE The total is not required at this stage	M1	1.1a
Obtains the correct probability AWFW [0.71, 0.713] Ignore subsequent incorrect simplification once correct fraction or decimals obtained	A1	1.1b
	(2)

Mark scheme (c)

Scheme	Marks	AO
Finds correct \(P(C) \times P(T)\) or uses correct conditional probability eg \(P(T \mid C) = \dfrac{21}{153}\) OE All figures must be correct	M1	3.1b
Compares \(P(C \cap T) = \dfrac{21}{240}\) with \(\dfrac{153}{1280}\) and concludes owning a cat and owning a tortoise are not independent eg \(\dfrac{21}{240} \ne \dfrac{153}{1280}\) so not independent or compares conditional probability eg \(\dfrac{21}{153} \ne \dfrac{45}{240}\) so not independent All figures must be correct	R1	2.2a
	(2)
	(10 marks)

Typical solution

\[P(C) \times P(T) = \frac{153}{240} \times \frac{45}{240} = \frac{153}{1280}\]\[= 0.12\]\[P(C \cap T) = \frac{21}{240}\]\[= 0.0875\]\[0.12 \ne 0.0875\]

Hence events \(C\) and \(T\) are not independent