June 2025 Paper 3 Q19

AQACurrent spec10 marksProbability

19 Three events \(X\), \(Y\) and \(Z\) are such that

\[\begin{gathered}\mathrm{P}(X) = \mathrm{P}(Y) = 0.38\\ \mathrm{P}(X \cup Z) = 0.5\\ \mathrm{P}[(X \cup Y)^\prime] = 0.35\end{gathered}\]

\(X\) and \(Z\) are mutually exclusive.

(a) Find \(\mathrm{P}(Z)\) [1 mark]

(b) Complete the Venn diagram.

Venn diagram in a rectangle: three circles X, Y and Z in a row, X overlapping Y and Y overlapping Z, X and Z not overlapping. X only is 0.27, outside all circles is 0.3, and the regions X and Y, Y only, Y and Z, and Z only are blank

[3 marks]

(d) Find \(\mathrm{P}(Y^\prime \mid Z^\prime)\) [2 marks]

(e) Show that the events \(Y\) and \(Z\) are not independent of each other. [2 marks]

Scheme	Marks	AO
Obtains 0.12	B1	1.1a
	(1)

Completed Venn diagram: X only 0.27, X and Y 0.11, Y only 0.2, Y and Z 0.07, Z only 0.05, outside 0.3 — Mark scheme (b)

Scheme	Marks	AO
Obtains at least one of 0.11 or 0.2 or 0.07 or 0.05 in the correct place	M1	1.1a
Obtains at least two of 0.11 or 0.2 or 0.07 or 0.05 in the correct place	M1	1.1a
Completes the Venn diagram correctly	A1	1.1b
	(3)

Venn diagram with the regions X and Y (0.11) and Y and Z (0.07) shaded — Mark scheme (c)

Scheme	Marks	AO
States their 0.11 + their 0.07 PI by correct answer	M1	1.1a
Obtains 0.18 FT their values provided the final positive answer is <0.38	A1F	1.1b
	(2)

Mark scheme (d)

Scheme	Marks	AO
States \(\mathrm{P}(Y^\prime \mid Z^\prime) = \dfrac{P(Y^\prime \cap Z^\prime)}{P(Z^\prime)}\) OE Condone missing \(\mathrm{P}(Y^\prime \mid Z^\prime)\) or calculates \(\mathrm{P}(Y^\prime \cap Z^\prime)\) PI by 0.57 seen on numerator or calculates 1 − their \(\mathrm{P}(Z)\) from part 19(a) PI by 0.88 seen on denominator PI by correct answer	M1	1.1a
Obtains \(\dfrac{57}{88}\) or AWFW [0.647, 0.65]	A1	1.1b
	(2)

Typical solution

\[\mathrm{P}(Y^\prime \mid Z^\prime) = \frac{P(Y^\prime \cap Z^\prime)}{P(Z^\prime)}\]\[= \frac{0.3 + 0.27}{1 - 0.12}\]\[= \frac{0.57}{0.88}\]\[= \frac{57}{88}\]

Mark scheme (e)

Scheme	Marks	AO
States their \(\mathrm{P}(Z)\) from part 19(a) \(\times\) 0.38 or compares their \(\mathrm{P}(Y \mid Z)\) with 0.38 or compares their \(\mathrm{P}(Z \mid Y)\) with their \(\mathrm{P}(Z)\) from part 19(a) or any other valid comparison with one correct probability to at least 2 sf	M1	1.1a
Completes a reasoned argument with correct values and concludes that \(Y\) and \(Z\) are not independent	R1	2.1
	(2)
	(10 marks)

Typical solution

\[\mathrm{P}(Y) \times \mathrm{P}(Z) = 0.38 \times 0.12\]\[= 0.0456\]\[P(Y \cap Z) = 0.07\]\[0.0456 \neq 0.07\]\[P(Y \cap Z) \neq \mathrm{P}(Y) \times \mathrm{P}(Z)\]

Hence events \(Y\) and \(Z\) are not independent.