18N.1.AHL.TZ0.H_1

pestleMathematicsAAHLPaper 118N· sl-4-6-combined-mutually-exclusive-conditional-independence-prob-diagramssource ↗

Consider two events, $A$ and $B$ , such that ${\text{P}}\left( A \right) = {\text{P}}\left( {A' \cap B} \right) = 0.4$ and ${\text{P}}\left( {A \cap B} \right) = 0.1$ .

By drawing a Venn diagram, or otherwise, find ${\text{P}}\left( {A \cup B} \right)$ .

[3]

Show that the events $A$ and $B$ are not independent.

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

(M1)

Note: Award M1 for a Venn diagram with at least one probability in the correct region.

EITHER

${\text{P}}\left( {A \cap B'} \right) = 0.3$ (A1)

${\text{P}}\left( {A \cup B} \right) = 0.3 + 0.4 + 0.1 = 0.8$ A1

${\text{P}}\left( B \right) = 0.5$ (A1)

${\text{P}}\left( {A \cup B} \right) = 0.5 + 0.4 - 0.1 = 0.8$ A1

[3 marks]

METHOD 1

${\text{P}}\left( A \right){\text{P}}\left( B \right) = 0.4 \times 0.5$ (M1)

= 0.2 A1

statement that their ${\text{P}}\left( A \right){\text{P}}\left( B \right) \ne {\text{P}}\left( {A \cap B} \right)$ R1

Note: Award R1 for correct reasoning from their value.

⇒ $A$ , $B$ not independent AG

METHOD 2

${\text{P}}\left( {\left. A \right|B} \right) = \frac{{{\text{P}}\left( {A \cap B} \right)}}{{{\text{P}}\left( B \right)}} = \frac{{0.1}}{{0.5}}$ (M1)

= 0.2 A1

statement that their ${\text{P}}\left( {\left. A \right|B} \right) \ne {\text{P}}\left( A \right)$ R1

Note: Award R1 for correct reasoning from their value.

⇒ $A$ , $B$ not independent AG

Note: Accept equivalent argument using ${\text{P}}\left( {\left. B \right|A} \right) = 0.25$ .

[3 marks]

Examiners’ report

[N/A]