22M.2.SL.TZ1.6

METHOD 1

EITHER

one of $\text{P}\left(A\right)=x+0.16$  OR  $\text{P}\left(B\right)=x+0.36$           A1

OR

           A1

 

THEN

attempt to equate their $\text{P}(A\cap B)$ with their expression for $\text{P}\left(A\right)\times \text{P}\left(B\right)$           M1

$\text{P}(A\cap B)=\text{P}\left(A\right)\times \text{P}\left(B\right)\Rightarrow x=\left(x+0.16\right)\times \left(x+0.36\right)$           A1

$x=0.24$           A1

 

METHOD 2

attempt to form at least one equation in $\text{P}\left(A\right)$ and $\text{P}\left(B\right)$ using independence           M1

$\left(\text{P}\left(A\cap B\text{'}\right)=\text{P}\left(A\right)\times P\left(B\text{'}\right)\Rightarrow \right) \text{P}\left(A\right)\times \left(1-\text{P}\left(B\right)\right)=0.16$  OR

$\left(\text{P}\left(A\text{'}\cap B\right)=\text{P}\left(A\text{'}\right)\times P\left(B\right)\Rightarrow \right) \left(1-\text{P}\left(A\right)\right)\times \text{P}\left(B\right)=0.36$

$\text{P}\left(A\right)=0.4$ and $\text{P}\left(B\right)=0.6$           A1

$\text{P}(A\cap B)=\text{P}\left(A\right)\times \text{P}\left(B\right)=0.4\times 0.6$            (A1)

$x=0.24$           A1

 

[4 marks]

METHOD 1

recognising $\text{P}\left(A\text{'} \overline{) B\text{'}}\right)=\text{P}\left(A\text{'}\right)$            (M1)

$=1-0.16-0.24$

$=0.6$           A1

 

METHOD 2

$\text{P}\left(B\right)=0.36+0.24\left(=0.6\right)$

$\text{P}\left(A\text{'} \overline{) B\text{'}}\right)=\frac{\text{P}\left(A\text{'}\cap B\text{'}\right)}{\text{P}\left(B\text{'}\right)} \left(=\frac{0.24}{0.4}\right)$            (A1)

$=0.6$           A1

 

[2 marks]

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