N2018 P2 Q7

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

1. Find an expression for \(\mathrm{P}\left( A'\cap B' \right)\) and hence prove that \(A'\) and \(B'\) are independent events.[2]
2. Find an expression for \(\mathrm{P}\left( A'\cap C' \right)\). Draw a Venn diagram to illustrate to case when \(A'\) and \(C'\) are also mutually exclusive events. (You should not show event \(B\)on your diagram.)[2]

You are now given that \(A'\) and \(C'\) are not mutually exclusive, \(P(A)=\frac{2}{5}\), \(\mathrm{P}\left( B\cap C \right)=\frac{1}{5}\) and \(\mathrm{P}\left( A'\cap B'\cap C' \right)=\frac{1}{10}\).

3. Find exactly the maximum and minimum possible values of \(\mathrm{P}\left( A\cap B \right)\).[4]