2024 MI H1 Q10

A group of students is asked whether they participate in running, cycling or swimming activities on a regular basis. The numbers of students participating in different combinations of these activities are shown in the above Venn diagram. The number of students participating in all of these activities and none of these activities is \(8\) and \(x\) respectively. One of the students is chosen at random.

\(R\) is the event that the student participates in running.\n\(C\) is the event that the student participates in cycling.\n\(S\) is the event that the student participates in swimming.

1. Write down expressions for \(\mathrm{P}(R)\) and \(\mathrm{P}(S)\) in terms of \(x\).[2]
2. Given that the events \(R\) and \(S\) are independent, show that \(x = 33\).[2]

You may use this value of \(x\) for the rest of this question.

3. Find \(\mathrm{P}(S \cap R)\).[1]
4. Find \(\mathrm{P}(C \cup R')\).[1]

Two students from the whole group are chosen at random.

5. Find the probability that each student participates in exactly two of these three activities (running, cycling, swimming).[2]