Question

Rhode Island currentine because of extensive vaccination, health related safety measures, and regular testing, responsible for COVID has relatively low active covid cases, say 5%. Suppose the test conducted by a company false positive results ( 19 PCR testing in Rhode Island is 96% sensitive. Also, assume that it provides 5% randomly selected. The i.e. it inaccurately provides positive results for not Covid person). Let a resident is

(a) Create appropriate notations for the events, sketch a tree diagram, and populate the respective probabilities. Identify the conditional probabilities related to sensitivity, specificity, false positive, and false negative test results. Write these probabilities clearly as a list or bullet point.
(b) What is the probability that the selected resident has positive Covid test results?
(c) What is the probability that the selected person has negative test results?
(d) If a randomly selected resident received positive result during testing, what is the probability that the individual actually has Covid?
(e) If a randomly selected resident received negative result during testing, what is the probability that the individual does not have Covid?
(f) Given that the resident received negative result during testing, what is the probability that the resident still has Covid?

Answer 1

In this question, we are given the sensitivity and specificity of a COVID test. We use these probabilities to calculate the probability of a positive or negative test result, the probability of having COVID given a positive or negative test result, and the probability of not having COVID given a negative test result.

Step-by-step explanation:

In this question, we are given the sensitivity and specificity of a COVID test. Let's define the following events:

• A: The resident has COVID
• P: The test is positive
• A': The resident does not have COVID
• N: The test is negative

(a) The tree diagram and probabilities can be constructed as follows:

Sensitivity (True positive rate) = P(P|A) = 0.96

Specificity (True negative rate) = P(N|A') = 0.95

False Positive Rate = P(P|A') = 0.05

False Negative Rate = P(N|A) = 0.04

(b) The probability that the resident has a positive COVID test result is P(P) = P(P|A) * P(A) + P(P|A') * P(A') = 0.96 * 0.05 + 0.05 * 0.95 = 0.049 + 0.0475 = 0.0965 (or 9.65%)

(c) The probability that the resident has a negative COVID test result is P(N) = P(N|A) * P(A) + P(N|A') * P(A') = 0.04 * 0.05 + 0.95 * 0.95 = 0.002 + 0.9025 = 0.9045 (or 90.45%)

(d) The probability that the resident actually has COVID given a positive test result is P(A|P) = P(P|A) * P(A) / P(P) = 0.96 * 0.05 / 0.0965 = 0.048 / 0.0965 = 0.497 (or 49.7%)

(e) The probability that the resident does not have COVID given a negative test result is P(A'|N) = P(N|A') * P(A') / P(N) = 0.95 * 0.95 / 0.9045 = 0.9025 / 0.9045 = 0.9978 (or 99.78%)

(f) The probability that the resident still has COVID given a negative test result is P(A|N) = 1 - P(A'|N) = 1 - 0.9978 = 0.0022 (or 0.22%)