Understand and use the notion of independence of an event 𝐴 from an event 𝐵, as defined by 𝑃(𝐴|𝐵) = 𝑃(𝐴).

A survey of 200 students asks whether they play cricket and whether they play tennis. The results show that 60 students play cricket, 80 students play tennis, and 24 students play both sports. Determine whether the events 'a student plays cricket' and 'a student plays tennis' are independent. Justify your answer using the definition of independence.

A market research company surveyed 500 shoppers about their preferences for online and in-store shopping. The results are shown in the table below. Determine whether the event 'a shopper prefers online shopping' is independent of the event 'a shopper is aged 18–35 years'. Show all working and justify your conclusion.

A university surveyed 180 students about their study habits and assignment completion status. The data collected is shown in the table below. Determine whether the event "Student attends lectures regularly" (event \(L\)) is independent of the event "Student submits assignments on time" (event \(A\)), using the definition \(P(L|A) = P(L)\). Show all working.