21M.2.SL.TZ2.4

EITHER

$\text{P}\left(S\right)+\text{P}\left(T\right)+\text{P}\left(S\text{'}\cap T\text{'}\right)-\text{P}\left(S\cap T\right)=1$  OR  $\text{P}\left(S\cup T\right)=\text{P}\left(\left(S\text{'}\cap T\text{'}\right)\mathrm{\text{'}}\right)$          (M1)

$0.7+0.2+0.18-\text{P}\left(S\cap T\right)=1$  OR  $\text{P}\left(S\cup T\right)=1-0.18$

OR

a clearly labelled Venn diagram        (M1)

THEN

$P\left(S\cap T\right)=0.08$  (accept $8\%$)              A1

 

Note: To obtain the M1 for the Venn diagram all labels must be correct and in the correct sections. For example, do not accept $0.7$ in the area corresponding to $S\cap T\text{'}$.

 

[2 marks]

EITHER

$\text{P}\left(T\cap S\text{'}\right)=\text{P}\left(T\right)-\text{P}\left(T\cap S\right)\left(=0.2-0.08\right)$  OR

$\text{P}\left(T\cap S\text{'}\right)=\text{P}\left(T\cup S\right)-\text{P}\left(S\right)\left(=0.82-0.7\right)$         (M1)

 
OR

a clearly labelled Venn diagram including $\text{P}\left(S\right)$, $\text{P}\left(T\right)$ and $\text{P}\left(S\cap T\right)$         (M1)

THEN

$=0.12$  (accept $12\%$)              A1

 

[2 marks]

$\text{P}\left(G\cap T\right)=\text{P}\left(T/G\right)\text{P}\left(G\right) \left(0.25\times 0.48\right)$         (M1)

$=0.12$             A1

 

[2 marks]

METHOD 1

$\text{P}\left(G\right)\times \text{P}\left(T\right)\left(=0.48\times 0.2\right)=0.096$             A1

$\text{P}\left(G\right)\times \text{P}\left(T\right)\mathrm{\ne }\text{P}\left(G\cap T\right)\mathrm{\Rightarrow }$  $G$ and $T$ are not independent             R1

 

METHOD 2

$\text{P}\left(T \overline{) G}\right)=0.25$             A1

$\text{P}\left(T \overline{) G}\right)\ne \text{P}\left(T\right)\mathrm{\Rightarrow }$  $G$ and $T$ are not independent             R1

 

Note: Do not award A0R1.

 

[2 marks]

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