IB MATH AI SL, Paper 2, May, 2021, TZ2, Solved Past Paper

A medical centre is conducting tests on patients to detect a specific disease that affects $5\mathrm{\%}$ of the population. The testing process involves a sampling method to select patients for testing.

The test is designed to yield a positive result if a patient has the disease and a negative result if they do not. However, the test is not flawless. It correctly identifies $99\mathrm{\%}$ of those with the disease as positive, but $2\mathrm{\%}$ of those without the disease also test positive. A tree diagram is used to represent this information visually.

Additionally, the medical centre staff analyzed the care provided to all patients visiting on a randomly chosen day. Each patient received at least one of the following services: medical tests (M), consultation with a nurse (N), or consultation with a doctor (D). The data collected includes:

• 78 patients had medical tests.
• 45 patients were seen by a nurse.
• 30 patients were seen by a doctor.
• 9 patients had medical tests and were seen by both a doctor and a nurse.
• 18 patients had medical tests and were seen by a doctor but not by a nurse.
• 11 patients were seen by a nurse and had medical tests but were not seen by a doctor.
• 2 patients were seen by a doctor without being seen by a nurse or having medical tests.

The task involves determining the sampling method used, analyzing the tree diagram to find specific probabilities, explaining discrepancies between expected and actual test results, illustrating the data with a Venn diagram, and calculating the total number of patients who visited the centre on that day.