The Gold Coast hospital is testing patients for COVID-19. If a patient has the disease, the test is designed to return a "positive" result. If a patient does not have the disease, the test should return a "negative" result. As we know, no test is perfect though. For this test, 99% of patients who have COVID-19 will test positive (i.e., the test has a 'false negative' rate of 1%). Also, 5% of patients who don't have COVID-19 will also test positive (i.e., the test has a 'false positive' rate of 5%). The actual percentage of patients with the disease is 10%.

1. (4 marks) Construct a tree diagram showing all possible cases and their probabilities, including the probabilities of all possible combinations of outcomes.
2. (4 marks) Determine the probability that a random patient who tests positive, has COVID-19. You must show your work - preferably showing the calculations based on the "rules of probability"