2023 JPJC P2 Q8

The events \(A\) and \(B\) are such that \(\mathrm{P}\left( A \right)=a\) and \(\mathrm{P}\left( B \right)=b\). \(A\) and \(B\) are mutually exclusive events.

1. Find an expression for \(\mathrm{P}\left( {A}'\cap {B}' \right)\).[1]
2. Draw a Venn diagram to illustrate the case where \({A}'\) and \({B}'\) are also mutually exclusive events. Hence, state an equation relating \(a\) and \(b\).[2]

For the rest of the question, \({A}'\) and \({B}'\) are not mutually exclusive events. The event \(C\) is such that \(\mathrm{P}\left( C \right)=c\). \(A\) and \(C\) are independent events. It is also given that \(a=0.2\), \(b=0.3\) and \(\mathrm{P}\left( B\cap C \right)=0.14\).

3. Find an expression for \(\mathrm{P}\left( A\cup B\cup C \right)\).[3]
4. Find an expression for \(\mathrm{P}\left( {A}'\cap {B}'\cap C \right)\).[1]
5. Use your answers to parts (iii) and (iv) to find the range of values of \(c\).[2]