Question

One day, you’re feeling a little under the weather. Since you’re a responsible human being, you go get tested for COVID. You want to go to a party that night, so you take a rapid test. You know that the rapid test has 80% sensitivity [P(+|infected) = 0.8] and a 60% specificity [P(-|infected) = 0.6]. You also know that about 50% of the people in your area who feel sick have COVID. The test comes back negative. Given this, what is the probability that you do, in fact, have COVID.

Answer 1

I'm not sure what your question is so I just assumed you were trying to find the probability of you having covid.

Explanation:

To determine the probability that you have COVID given a negative rapid test result, we can use Bayes' theorem. Bayes' theorem allows us to update the probability of an event based on new information.

Let's define the following probabilities:

P(C) = Probability of having COVID

P(N) = Probability of a negative test result

P(+|C) = Sensitivity of the test (Probability of a positive result given that you have COVID)

P(-|C) = 1 - P(+|C) = False negative rate (Probability of a negative result given that you have COVID)

P(-|non-C) = Specificity of the test (Probability of a negative result given that you don't have COVID)

P(non-C) = Probability of not having COVID

Based on the information provided:

P(+|C) = 0.8 (sensitivity)

P(-|C) = 1 - P(+|C) = 0.2 (false negative rate)

P(-|non-C) = 0.6 (specificity)

P(non-C) = 1 - P(C) = 1 - 0.5 = 0.5 (probability of not having COVID)

Now, let's calculate the probability of having COVID given a negative test result using Bayes' theorem:

P(C|N) = (P(N|C) * P(C)) / P(N)

P(N|C) = P(-|C) * P(C) = 0.2 * 0.5 = 0.1 (probability of a negative result given that you have COVID)

P(N) = P(-|C) * P(C) + P(-|non-C) * P(non-C) = 0.2 * 0.5 + 0.6 * 0.5 = 0.1 + 0.3 = 0.4 (probability of a negative test result)

Plugging these values into Bayes' theorem:

P(C|N) = (0.1 * 0.5) / 0.4 = 0.05 / 0.4 = 0.125

Therefore, the probability that you have COVID given a negative rapid test result is 0.125 or 12.5%.