2024 SAJC H1 Q7

A group of students are surveyed on the types of game consoles, namely, Ybox, ClayStation and Mintendo Switch, that they owned. The number of students owning different combinations of game consoles are shown in the above Venn diagram. The number of students owning all three game consoles is \(x\). One student is chosen at random.

\(Y\) is the event that the student owns a Ybox
\(C\) is the event that the student owns a ClayStation

\(M\) is the event that the student owns a Mintendo Switch

1. Write down expressions for \(\mathrm{P}(Y)\) and \(\mathrm{P}(C)\) in terms of \(x\).[2]
2. Given that the events \(Y\) and \(C\) are independent, find the value of \(x\).[3]
3. Explain in the context of this question, what is meant by \(\mathrm{P}(M|Y)\), and find this value[2]

Three students are chosen at random, without replacement.

4. Find the probability that each student owns exactly two out of the three types of game consoles.[2]