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Date	May 2017	Marks available	3	Reference code	17M.2.hl.TZ1.1
Level	HL only	Paper	2	Time zone	TZ1
Command term	Find	Question number	1	Adapted from	N/A

Question

Consider two events $A$ and $B$ such that ${\text{P}}(A) = k,{\text{ P}}(B) = 3k,{\text{ P}}(A \cap B) = {k^2}$ and ${\text{P}}(A \cup B) = 0.5$ .

Calculate $k$ ;

[3]

a.

Find ${\text{P}}(A' \cap B)$ .

[3]

b.

Markscheme

use of ${\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B)$ M1

$0.5 = k + 3k - {k^2}$ A1

${k^2} - 4k + 0.5 = 0$

$k = 0.129$ A1

Note: Do not award the final A1 if two solutions are given.

[3 marks]

a.

use of ${\text{P}}(A' \cap B) = {\text{P}}(B) - {\text{P}}(A \cap B)$ or alternative (M1)

${\text{P}}(A' \cap B) = 3k - {k^2}$ (A1)

$= 0.371$ A1

[3 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

Topic 5 - Core: Statistics and probability » 5.3 » Combined events; the formula for

$P\left( {A \cup B} \right)$ .

17N.2.hl.TZ0.2c: Hence show that events $A’$ and $B$ are independent.
17N.2.hl.TZ0.2b: Find ${\text{P}}(A)$ .
17N.2.hl.TZ0.2a: Find ${\text{P}}(B)$ .
17M.2.hl.TZ1.1a: Calculate $k$ ;
13M.1.hl.TZ1.9: Two events A and B are such that ${\text{P}}(A \cup B) = 0.7$ and...
11M.1.hl.TZ1.1a: Find the value of ${\text{P}}(A \cup B)$ when(i) $A$ and $B$ are mutually...
14M.1.hl.TZ2.1: Events $A$ and $B$ are such that...
15M.1.hl.TZ2.1a: Find ${\text{P}}(A \cap B)$ .
15N.2.hl.TZ0.1a: Find ${\text{P}}(A \cap B)$ .
14N.1.hl.TZ0.4a: Determine the value of ${\text{P}}(A \cup B)$ when (i) $A$ and $B$ are mutually...