2024 CJC P2 Q10
2024 CJC P2 Q10
13 marks
Prelims
- In a certain university, there are \(110\) Co-Curricular Activities (CCAs) clustered into \(4\) categories. There are \(22\) Arts and Culture CCAs, \(16\) Community Service CCAs, \(38\) Physical Sports CCAs and \(34\) Special Interest CCAs.
- Albert wishes to find out about approaches to training of the Physical Sports CCAs, so he sends a questionnaire to \(22\) Physical Sports CCAs. Explain whether these \(22\) Physical Sports CCAs form a sample or a population.[1]
- Benedict wishes to investigate the level of student engagement in CCAs, but does not want to obtain the detailed information necessary from all \(110\) CCAs. Explain how he should carry out his investigation, and why he should do the investigation in this way.[2]
- Find the number of different possible samples of \(16\) CCAs, with \(4\) CCAs chosen from each category.[2]
- The probability of events \(A\), \(B\) and \(C\) occurring are given by \(\mathrm{P}\left( A \right)=\frac{8}{25}\), \(\mathrm{P}\left( B \right)=\frac{61}{100}\) and \(\mathrm{P}\left( C \right)=\frac{7}{20}\) respectively. It is also known that \(\mathrm{P}\left( A\cap C \right)=\frac{11}{100}\), \(\mathrm{P}\left( \left. B \right|C \right)=\frac{13}{35}\) and \(\mathrm{P}\left( A\cap B\cap C \right)=\frac{7}{100}\).
- Find \(\mathrm{P}\left( B\cap C \right)\).[2]
- If events \(A\) and \(B\) are independent, find \(\mathrm{P}\left( A\cap B\cap C' \right)\).[2]
- If events \(A\) and \(B\) are not known to be independent, find the greatest and least possible values of \(\mathrm{P}\left( A\cap B\cap C' \right)\).[4]
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Answer:(a)(i) These \(22\) CCAs form a sample since this number is less than the population size of Physical Sports CCAs which is \(38\). (ii) Benedict should select a random sample from the 110 CCAs: every CCA has an equal probability to be selected to be in the sample and the CCAS are selected independently. A random sample helps to avoid bias. (iii) \(4.56\times {{10}^{16}}\) (\(3\) s.f.) (b)(i) \(\frac{13}{100}\) (ii) \(\frac{313}{2500}\)
(iii) Minimum \(\text{P}\hspace{0.1em}\left( A\cap B\cap C' \right)=\frac{1}{25}\) and maximum \(\text{P}\hspace{0.1em}\left( A\cap B\cap C' \right)=\frac{21}{100}\).
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