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Part B Many of us use rapid antigen tests to assess whether or not we have COVID-19. The false positive rate of a rapid antigen test is the probability that the antigen test returns a positive result given that you do NOT have COVID-19. So, if + is the event that the test returns a positive result and C is the event that you have COVID-19, the false positive rate is p(+|C). The false negative rate of a rapid antigen test is p(-|C), the event that the test incorrectly returns a negative result when you do in fact have COVID-19. The true positive rate is p(+|C) and the true negative rate is p(-|C). Popular rapid antigen tests have a true positive rate for symptomatic individuals of approximately 73% and a true negative rate of approximately 99.6%.¹ The false positive rate is approximately 0.4% and the false negative rate is approximately 27%. Suppose that you wake up one morning with the sniffles. Because you are very responsible, you take a rapid antigen test. 1. The test comes up negative. What is the probability that you have COVID-19, given that you have a negative test and the sniffles? This is p(C – S). Hints: o The way to approach this is to use Bayes' rule to write p(NSC)p(C) p(C| - ns) = p(NSC)p(C)+p(-¯S|Ñ')p(T) . Then, compute each of the terms appearing in this fraction. Because you always have the sniffles when you have COVID-19, p(- n SC) = P(-|C). o You also need to calculate p(— ʼn S|¯)p(C). You may assume that p(− n S|C) = p(−|C)p(SC). This is called conditional independence, an important topic which we sadly won't hvae time to discuss. 2. The test comes up positive. What is the probability that you have COVID-19, given that you have a positive test and the sniffles? o This problem is very similar to the previous one. You may assume that p(+nSC) = p(+C)p(SC).

Part B Many of us use rapid antigen tests to assess whether or not we have COVID-19. The false positive rate of a rapid antigen test is the probability that the antigen test returns a positive result given that you do NOT have COVID-19. So, if + is the event that the test returns a positive result and C is the event that you have COVID-19, the false positive rate is p(+|C). The false negative rate of a rapid antigen test is p(-|C), the event that the test incorrectly returns a negative result when you do in fact have COVID-19. The true positive rate is p(+|C) and the true negative rate is p(-|C). Popular rapid antigen tests have a true positive rate for symptomatic individuals of approximately 73% and a true negative rate of approximately 99.6%.¹ The false positive rate is approximately 0.4% and the false negative rate is approximately 27%. Suppose that you wake up one morning with the sniffles. Because you are very responsible, you take a rapid antigen test. 1. The test comes up negative. What is the probability that you have COVID-19, given that you have a negative test and the sniffles? This is p(C – S). Hints: o The way to approach this is to use Bayes' rule to write p(NSC)p(C) p(C| - ns) = p(NSC)p(C)+p(-¯S|Ñ')p(T) . Then, compute each of the terms appearing in this fraction. Because you always have the sniffles when you have COVID-19, p(- n SC) = P(-|C). o You also need to calculate p(— ʼn S|¯)p(C). You may assume that p(− n S|C) = p(−|C)p(SC). This is called conditional independence, an important topic which we sadly won't hvae time to discuss. 2. The test comes up positive. What is the probability that you have COVID-19, given that you have a positive test and the sniffles? o This problem is very similar to the previous one. You may assume that p(+nSC) = p(+C)p(SC).

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Part B Many of us use rapid antigen tests to assess whether or not we have COVID-19. The false positive rate of a rapid antigen test is the probability that the antigen test returns a positive result given that you do NOT have COVID-19. So, if + is the event that the test returns a positive result and C is the event that you have COVID-19, the false positive rate is p(+|C). The false negative rate of a rapid antigen test is p(-|C), the event that the test incorrectly returns a negative result when you do in fact have COVID-19. The true positive rate is p(+|C) and the true negative rate is p(-|C). Popular rapid antigen tests have a true positive rate for symptomatic individuals of approximately 73% and a true negative rate of approximately 99.6%.¹ The false positive rate is approximately 0.4% and the false negative rate is approximately 27%. Suppose that you wake up one morning with the sniffles. Because you are very responsible, you take a rapid antigen test. 1. The test comes up negative. What is the probability that you have COVID-19, given that you have a negative test and the sniffles? This is p(C – S). Hints: o The way to approach this is to use Bayes' rule to write p(NSC)p(C) p(C| - ns) = p(NSC)p(C)+p(-¯S|Ñ')p(T) . Then, compute each of the terms appearing in this fraction. Because you always have the sniffles when you have COVID-19, p(- n SC) = P(-|C). o You also need to calculate p(— ʼn S|¯)p(C). You may assume that p(− n S|C) = p(−|C)p(SC). This is called conditional independence, an important topic which we sadly won't hvae time to discuss. 2. The test comes up positive. What is the probability that you have COVID-19, given that you have a positive test and the sniffles? o This problem is very similar to the previous one. You may assume that p(+nSC) = p(+C)p(SC).
Transcribed Image Text:Part B Many of us use rapid antigen tests to assess whether or not we have COVID-19. The false positive rate of a rapid antigen test is the probability that the antigen test returns a positive result given that you do NOT have COVID-19. So, if + is the event that the test returns a positive result and C is the event that you have COVID-19, the false positive rate is p(+|C). The false negative rate of a rapid antigen test is p(-|C), the event that the test incorrectly returns a negative result when you do in fact have COVID-19. The true positive rate is p(+|C) and the true negative rate is p(-|C). Popular rapid antigen tests have a true positive rate for symptomatic individuals of approximately 73% and a true negative rate of approximately 99.6%.¹ The false positive rate is approximately 0.4% and the false negative rate is approximately 27%. Suppose that you wake up one morning with the sniffles. Because you are very responsible, you take a rapid antigen test. 1. The test comes up negative. What is the probability that you have COVID-19, given that you have a negative test and the sniffles? This is p(C – S). Hints: o The way to approach this is to use Bayes' rule to write p(NSC)p(C) p(C| - ns) = p(NSC)p(C)+p(-¯S|Ñ')p(T) . Then, compute each of the terms appearing in this fraction. Because you always have the sniffles when you have COVID-19, p(- n SC) = P(-|C). o You also need to calculate p(— ʼn S|¯)p(C). You may assume that p(− n S|C) = p(−|C)p(SC). This is called conditional independence, an important topic which we sadly won't hvae time to discuss. 2. The test comes up positive. What is the probability that you have COVID-19, given that you have a positive test and the sniffles? o This problem is very similar to the previous one. You may assume that p(+nSC) = p(+C)p(SC).
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