Use set language and notation for events, including 𝐴 or 𝐴′ for the complement of an event 𝐴, 𝐴 ∩ 𝐵 for the intersection of event 𝐴 and event 𝐵, and 𝐴 ∪ 𝐵 for the union of event 𝐴 and event 𝐵, and recognise mutually exclusive events.

A student surveys 200 people about their music preferences. Let R be the event that a person likes rock music, and J be the event that a person likes jazz music. It is known that 78 people like rock, 52 people like jazz, and 15 people like both rock and jazz. (a) Calculate the number of people who like rock or jazz (or both). Express your answer using set notation. (b) State whether events R and J are mutually exclusive. Justify your answer.

A quality control inspector tests electronic components from two production lines, Line X and Line Y. Let event \(D\) be 'the component is defective' and event \(Y\) be 'the component is from Line Y'. The inspector records the following probabilities: \(P(D) = 0.12\), \(P(Y) = 0.65\), and \(P(D \cap Y) = 0.05\). (a) Use set notation to express the event 'the component is not defective and is from Line Y', then calculate its probability. [2 marks] (b) Determine whether the events \(D'\) and \(Y\) are mutually exclusive. Justify your answer using appropriate set notation and probability values. [2 marks]

A student rolls a standard six-sided die. Let \(A\) be the event 'the result is an even number' and let \(B\) be the event 'the result is a multiple of 3'. Which of the following events is mutually exclusive to \(A \cup B\)?

A survey of $180$ students asks about participation in sports. Let $S$ denote the event that a student plays soccer, and $T$ denote the event that a student plays tennis. The survey finds that $65$ students play soccer, $48$ students play tennis, and $22$ students play both sports. (a) Use set notation to write down the event that a student plays soccer or tennis (or both). [1] (b) Determine the number of students who play soccer or tennis. [1] (c) State whether the events $S$ and $T$ are mutually exclusive, and justify your answer using the survey data. [1]