2015 TPJC P2 Q5

Two fair dice are thrown simultaneously. Events \(A\) and \(B\) are defined as follows.

\(A\): \(\quad\) at least one six appears,

\(B\): \(\quad\) the sum of two scores is ten.

Find

1. \(\mathrm{P}\left( A|B \right)\),[2]
2. \(\mathrm{P}\left( A|B' \right)\).[2]

For a third event \(C\), it is given that \(\mathrm{P}\left( B'\cup C \right)=\frac{35}{36}\) and that \(B\) and \(C\) are independent.

3. Find the exact value of \(\mathrm{P}\left( C \right)\).[3]