Use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events.

A survey of 240 students recorded whether they participate in sport (S) and whether they study music (M). The data collected are shown in the table below. (a) Use the data to estimate the conditional probability that a student studies music given that they participate in sport. Give your answer as a decimal correct to 3 decimal places. [1 mark] (b) Use the data to estimate \(P(M)\), the probability that a randomly selected student studies music. Give your answer as a decimal correct to 3 decimal places. [1 mark] (c) Are the events S and M independent? Justify your answer by comparing appropriate probabilities. [1 mark]

A health clinic surveyed 2,400 patients about their exercise habits and cholesterol levels. The results are summarised in the table below. (a) Use the data to determine the probability that a randomly selected patient from this survey has high cholesterol, given that they exercise regularly. (1 mark) (b) Determine the probability that a randomly selected patient exercises regularly. (1 mark) (c) Use your answers from parts (a) and (b) to determine whether the events 'high cholesterol' and 'regular exercise' are independent for this data set. Justify your conclusion with appropriate calculations. (2 marks)

A sports centre surveyed 450 members about their preferred activity. The results are shown in the table below. Use the data to determine whether the events 'prefers fitness classes' and 'member aged under 30' appear to be independent. Show all working.