When the first Covid-19 tests were being used, there were a few issues with accuracy. In April of 2020, 99% of the people tested for Covid-19 did not have the virus. When they were tested, 1% of the tests showed a false positive response, and 30 percent of the tests showed a false negative response regardless of whether the person had Covid19 or not.

Question

When the first Covid-19 tests were being used, there were a few issues with accuracy. In April of 2020, 99% of the people tested for Covid-19 did not have the virus. When they were tested, 1% of the tests showed a false positive response, and 30 percent of the tests showed a false negative response regardless of whether the person had Covid19 or not.

show work:
1)what is the probability that a person tested positive for Covid19, if they had the virus?
2)what is the probability that a person who tested negative had Covid19?
3)what is the probability that someone tested negative?

Answer 1

To answer these questions, we need to use conditional probabilities. We can use the following information:

- P(positive test | has the virus) = 1 - P(false negative)
- P(negative test | does not have the virus) = 1 - P(false positive)
- P(positive test) = P(positive test | has the virus) * P(has the virus) + P(positive test | does not have the virus) * P(does not have the virus)
- P(negative test) = P(negative test | does not have the virus) * P(does not have the virus) + P(negative test | has the virus) * P(has the virus)

Let's calculate the probabilities:

1) Probability that a person tested positive for Covid19, given that they had the virus:

P(positive test | has the virus) = 1 - P(false negative) = 1 - 0.30 = 0.70

2) Probability that a person who tested negative had Covid19:

P(has the virus | negative test) = (P(negative test | has the virus) * P(has the virus)) / P(negative test)

We need to calculate P(negative test) first:

P(negative test) = P(negative test | does not have the virus) * P(does not have the virus) + P(negative test | has the virus) * P(has the virus)
= 0.99 * (1 - P(false positive)) + 0.01 * P(has the virus)
= 0.99 * (1 - 0.01) + 0.01 * P(has the virus)
= 0.99 * 0.99 + 0.01 * P(has the virus)

Now we can calculate P(has the virus | negative test):

P(has the virus | negative test) = (P(negative test | has the virus) * P(has the virus)) / P(negative test)
= (0.01 * P(has the virus)) / (0.99 * 0.99 + 0.01 * P(has the virus))

3) Probability that someone tested negative:

P(negative test) = P(negative test | does not have the virus) * P(does not have the virus) + P(negative test | has the virus) * P(has the virus)
= 0.99 * (1 - P(false positive)) + 0.01 * P(has the virus)
= 0.99 * (1 - 0.01) + 0.01 * P(has the virus)
= 0.99 * 0.99 + 0.01 * P(has the virus)

Now you can use these calculations to find the exact probabilities based on the given information.