Question 2. [14 marks] To get full marks for the following questions you need to be extremely clear in what you are commu- nicating. The best way to do this is to use mathematical notation. This means clearly defining your events where necessary, and using correct probability statements. After calculating probabili- ties, provide your final answer in a sentence using plain English. Very few testing procedures are absolutely perfect. This has become particularly apparent with the discussion of sensitivity and specificity of COVID-19 Rapid Antigen (RAT) tests recently. These have often been compared to the gold standard Polymerase Chain Reaction (PCR) test. The information booklet for a high quality RAT test provides details for sensitivity and specificity. RAT tests were applied to 101 subjects who where known to have COVID-19. Of those, 91 resulted in a positive RAT test. Another 305 subjects who were known not to be infected with COVID-19 were tested using the RAT test and 303 of them tested negative. Looking on the NSW Health Website, there are currently 166,957 active cases of COVID-19 in NSW out of a population of 8,160,000 people. Let . be the event that someone has COVID-19 and therefore is the event that someone does not have COVID-19. Let A be the event that someone tests positive for COVID-19 and therefore Pc is the event that someone tests negative for COVID-19. (a) [1 mark] What is the probability that a randomly selected individual from NSW has COVID- 197 (b) [1 mark] Given someone has COVID-19, find the probability that a test will yield a positive result. (This probability represents the sensitivity of the test.) (c) [1 mark] Given someone does not have COVID-19, find the probability that the test will yield a negative result. (This probability represents the specificity of the test.) (d) [3 marks] Construct a tree diagram for the above scenario. Clearly label the arms representing sensitivity and specificity. (e) [3 marks] Given that a random individual tests positive for COVID-19 using a RAT test, find the probability that the individual actually has the disease. (This probability represents the positive predictive value of the test and is what we would actually care about if we were tested.) (f) [1 mark] Comment on the relative size of your probability in part (e), taking into account the fact that this is a high quality RAT test with very good sensitivity and specificity. (g) [3 marks] Suppose you went to a family birthday party on the weekend and there were 10 individuals there. On Wednesday following the party, 6 out of 10 people at the party have tested positive for COVID-19 using a PCR test. Now suppose that you test positive using a RAT test. Estimate the probability of having COVID-19 given your positive RAT test taking into account the information you received following the weekend's party. (h) [1 mark] Comment on how realistic the value is for the probability of someone having COVID- 19 from part (a)