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

[AI-generated answer — Explain Bot]

To answer these questions, we can use conditional probability. Let's define some variables first:

Let:
A be the event that a person tested positive for Covid-19
B be the event that a person had Covid-19

Given information:
P(B) = 1% (since 99% of people tested did not have the virus)
P(A|B) = 1 - 30% = 70% (1% false positive rate)
P(A|B') = 30% (30% false negative rate)

1) What is the probability that a person tested positive for Covid-19, if they had the virus?

To find P(A|B), we can use Bayes' theorem:

P(A|B) = (P(B|A) * P(A)) / P(B)

P(B|A) represents the probability that a person had Covid-19 given that they tested positive. Since P(B|A) = 1, we have:

P(A|B) = (1 * P(A)) / P(B)

We know that P(B) = 1%, so substituting the values:

P(A|B) = (1 * P(A)) / 0.01

We also know that P(A|B) = 70%, so:

0.7 = (1 * P(A)) / 0.01

Simplifying:

0.7 * 0.01 = P(A)

P(A) = 0.007 or 0.7%

Therefore, there is a 0.7% probability that a person tested positive for Covid-19 if they had the virus.

2) What is the probability that a person who tested negative had Covid-19?

To find P(B|A'), that is the probability that a person had Covid-19 given that they tested negative, we can again use Bayes' theorem:

P(B|A') = (P(A'|B) * P(B)) / P(A')

P(A'|B) represents the probability that a person tested negative given that they had Covid-19. Since P(A'|B) = 30%, we have:

P(B|A') = (0.3 * P(B)) / P(A')

We know that P(B) = 1% and P(A') is the complement of P(A), so:

P(A') = 1 - P(A) = 1 - 0.007 = 0.993

Substituting the values:

P(B|A') = (0.3 * 0.01) / 0.993

Simplifying:

0.003 = P(B|A')

Therefore, there is a 0.3% probability that a person who tested negative had Covid-19.

3) What is the probability that someone tested negative?

To find P(A'), the probability that someone tested negative, we can use complement rule of probability:

P(A') = 1 - P(A)

P(A) = 0.007 (probability of testing positive)

P(A') = 1 - 0.007

P(A') = 0.993

Therefore, there is a 99.3% probability that someone tested negative.