2024 RI H1 Q10

In a school's annual sports carnival, \(3\) main sports are played: Badminton, Football and Tennis. The Venn diagram below shows the number of students in a particular class who signed up for the \(3\) sports.

Venn diagram showing student participation in Badminton, Football, and Tennis

\(B\) is the event that the student signed up for Badminton.\n\(F\) is the event that the student signed up for Football.\n\(T\) is the event that the student signed up for Tennis.

1. Explain in context what \(\mathrm{P}(F \cap (B \cup T)')\) represents, and state its value in terms of \(x\).[2]
2. Given that events \(B\) and \(T\) are independent, show that \(x = 4\).[3]
3. Given that a student signed up for Football, find the probability that the student also signed up for Tennis.[2]
4. Two students are chosen at random without replacement. Find the probability that both students signed up for exactly \(2\) sports each.[3]