a)(i) $P$(final score is $12$) $ = P\left( {6,{\text{ }}6} \right) = 1/36$

(ii) $P\left[ {\left( {1,5} \right) + \left( {1,4} \right) + \left( {2,3} \right) + \left( {3,2} \right) + \left( {4,1} \right)} \right]$

$ = 5/36$

b) $P\left( A \right) = 1/6$

$P\left( B \right) = P\left[ {\left( {1,5} \right) + \left( {2,4} \right) + \left( {3,3} \right) + \left( {4,{\text{ }}2} \right) + \left( {5,1} \right)} \right]$

$ = 5/36$

$P\left( C \right) = 1 - P\left( {O,{\text{ }}O} \right) = 3/4$

$P$($A$ and $B$) $ = P$($1$ and $5$) $ = 1/36$

$ \ne P\left( A \right) \times P\left( B \right)$

$P$($A$ and $C$) $ = P\left[ {\left( {2,5} \right) + \left( {4,5} \right) + \left( {6,5} \right)} \right] = 3/36$

$ \ne P\left( B \right) \times P\left( C \right)$

None are independent.