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.

(d) [3 marks] Construct a tree diagram for the above scenario. Clearly label the arms representing sensitivity and specificity. P -P(C-P) . 0 25 184 sensitivity 0-225 C P (c ) . 0.25 . 0 915 subject 8((* ):0.75 PP(C'MP) . 0.75 x091 = 0.0073 C (10)=09 9 Cp(ctmp ) = 0.75-049 = 0 7% 25 (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.) P(CnP) P(C P) = P (P) 0.225 0.225+0.0075 = 0.9677 Hence, if an individual test positive for COVID-19 using a RAT test there is a 96.77% chance they truly have the disease.Question 2. [14 marks] To get full marks for the following questions you need to be extremely clear in what you are communicating. 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 probabilities, 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 were 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