2024 ACJC P2 Q6
2024 ACJC P2 Q6
8 marks
Prelims
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 independent events.
- Find an expression for \(\mathrm{P}\left( {A}'\cap {B}' \right)\) in terms of \(a\) and \(b\), and hence prove that \({A}'\) and \({B}'\) are independent events.[2]
It is given that \(\mathrm{P}\left( A\mid {B}' \right)=0.85\) and \({\mathrm{P}\left( {{B}'} \right)=0.8}\).
- Find \(\mathrm{P}\left( A\cap {B}' \right)\).[2]
For a third event \(C\), it is given that \(A\) and \(C\) are mutually exclusive and \(\mathrm{P}\left( {A}'\cap {C}' \right)=0.52\).
- Find \(\mathrm{P}\left( C \right)\).[1]
- Hence find the set of possible values of \(\mathrm{P}\left( {A}'\cap {B}'\cap {C}' \right)\).[3]
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Answer:(a) Since \({\mathrm{P}(A' \cap B') = \mathrm{P}(A')\mathrm{P}(B')}\), hence \({A'}\) and \({B'}\) are independent events. (b) \({0.12}\) (c) \({0.33}\) (d) \({0.35 \leq \mathrm{P}(A' \cap B' \cap C') \leq 0.52}\)
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