Understand the notion of a conditional probability and recognise and use language that indicates conditionality.

A medical clinic administers a screening test for a rare disease. Medical records show that 2% of the population actually has the disease. The test correctly identifies 95% of people who have the disease (true positive rate) and correctly identifies 98% of people who do not have the disease (true negative rate). If a patient tests positive, which statement correctly describes the conditional probability $P(\text{disease} \mid \text{test positive})$?

A student randomly selects a marble from a bag containing 12 red marbles, 8 blue marbles, and 5 green marbles. Given that the marble selected is not green, determine the probability that the marble is red.

A quality control inspector tests electronic components. From historical data, it is known that \(8\%\) of components have a manufacturing defect. The inspector uses a testing device that correctly identifies a defective component \(95\%\) of the time, and correctly identifies a non-defective component \(92\%\) of the time. A component is randomly selected and the testing device indicates that it is defective. Determine the probability that the component is actually defective, given that the testing device indicates it is defective. Express your answer as a fraction in simplified form or as a decimal correct to four decimal places.

A fitness centre recorded data on membership type and gym usage frequency for 120 members. Of the 120 members, 75 hold a standard membership. Among those with a standard membership, 60 members visit the gym at least three times per week. Among the 45 members with a premium membership, 36 members visit the gym at least three times per week. Determine the probability that a member visits the gym at least three times per week, given that the member holds a standard membership.