2024 TMJC H1 Q8

An owner of a bubble tea shop would like to introduce three new flavours of drinks: "Amazing", "Bubbly" and "Cheerful". A group of students are each given the three new flavoured drinks to sample and asked on whether they like the flavours. The numbers of students who like the different combinations of drinks are shown in the above Venn diagram. The number who do not like any of the flavours is \(x\). One of the students is chosen at random.

\(A\) is the event that the student likes "Amazing"
\(B\) is the event that the student likes "Bubbly"
\(C\) is the event that the student likes "Cheerful"

1. Given \(\mathrm{P}(A' | B') = \frac{1}{2}\), show that \(x = 12\).[3]
2. Determine whether the events \(A\) and \(B\) are independent.[2]
3. Find \(\mathrm{P}(A \cup C)\).[1]

Two students are chosen at random, without replacement.

4. Find the probability that exactly one student likes the drink "Amazing".[2]