values Interpret conditional and unconditional probabilities and expected Identify coherent probability models involving events and random variables Apply common probability models in real contexts Solve probability problems using mathematical properties and tools Choose a disease or medical condition that is interesting or relevant to you personally and for which a diagnostic tests is available (like the nuchal translucency test for Down Syndrome from the class example.) (You can choose COVID-19 if you want, but I hope not everyone does.) Note: you're not required to know all the terms like sensitivity and specificity below. but you do hear them in the news occasionally. a. Find an estimate of the probability that a person truly has the disease. This is called the prevalence or incidence of the disease. Cite the source (and provide a link) and describe briefly the data upon which the estimate is based. Write a sentence interpreting this probability in context. b. Find an estimate of the probability of testing positive for someone who truly has the disease. This is called the sensitivity of the test. (Careful: (100% - sensitivity) might be reported instead.) Cite the source (and provide a link) and describe briefly the data upon which the estimate is based. Write a sentence interpreting this probability in context. c. Find an estimate of the probability of testing negative for someone who truly does not have the disease. This is called the specificity of the test. (Careful: (100% - specificity) might be reported instead.) Cite the source (and provide a link) and describe briefly the data upon which the estimate is based. Write a sentence interpreting this probability in context. d. Construct a two-way table split by the test result (positive or negative) and disease status (truly has the disease or truly does not have the disease). Assume a nice round number total like 100,000. Fill in all the counts in the table. e. Use the table to estimate the probability that a person who tests positive truly has the disease. This is called the precision or positive predictive value. Write a sentence interpreting this probability in context. f. Use the table to estimate the probability that a person who tests positive truly does not have the disease. This is called the false discovery rate. Write a sentence interpreting this probability in context. g. Is a person who tests positive more likely to truly have the disease or not? How many times more likely? h. Suppose a randomly selected person takes the test. In which case is the person more likely to truly have the disease: if the test result is unknown, or if the test is positive? How many times more likely? i. Someone you know is given this test as part of a routine screening, and it comes back positive. The person is unaware if they have any of the risk factors of the disease. The person knows that you have taken probability and asks for your advice regarding their chances that they actually have the disease. What would you tell them? Write a few sentences explaining, as non-technically as possible. what you would tell the person. What are the main numbers they should pay attention to, and how should they interpret them? Your goal is to give the person the most accurate and relevant assessment without overwhelming them with numbers and details. (Of course, your probability assessment should not replace expert medical opinion, but maybe your advice can help the person while they wait for a follow up test.)