Last year, 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?
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To calculate the probabilities, we can use conditional probability formulas. Let's assign the following probabilities:
P(V) = Probability of having the virus = 1%
P(NV) = Probability of not having the virus = 99%
P(Pos|V) = Probability of testing positive given that the person has the virus = 100% (no false negatives)
P(Pos|NV) = Probability of testing positive given that the person does not have the virus = 1% (false positive rate)
P(Neg|NV) = Probability of testing negative given that the person does not have the virus = 100% (no false positives)
P(Neg|V) = Probability of testing negative given that the person has the virus = 30% (false negative rate)
Now, let's calculate the probabilities step by step:
1) Probability that a person tested positive for Covid19, if they had the virus (P(V|Pos)):
Using Bayes' theorem, we can calculate it as:
P(V|Pos) = [P(Pos|V) * P(V)] / [P(Pos|V) * P(V) + P(Pos|NV) * P(NV)]
Substituting the given values:
P(V|Pos) = [(1% * 100%)] / [(1% * 100%) + (1% * 99%)]
P(V|Pos) = 1% / (1% + 0.99%)
P(V|Pos) = 1% / 1.99%
Therefore, the probability that a person tested positive for Covid19, if they had the virus, is approximately 0.50 or 50%.
2) Probability that a person who tested negative had Covid19 (P(V|Neg)):
Using Bayes' theorem, we can calculate it as:
P(V|Neg) = [P(Neg|V) * P(V)] / [P(Neg|V) * P(V) + P(Neg|NV) * P(NV)]
Substituting the given values:
P(V|Neg) = [(30% * 1%)] / [(30% * 1%) + (100% * 99%)]
P(V|Neg) = 0.30% / (0.30% + 99%)
P(V|Neg) = 0.30% / 99.30%
Therefore, the probability that a person who tested negative had Covid19 is approximately 0.30% or 0.003.
3) Probability that someone tested negative (P(Neg)):
P(Neg) = P(Neg|V) * P(V) + P(Neg|NV) * P(NV)
P(Neg) = 30% * 1% + 100% * 99%
P(Neg) = 0.30% + 99%
P(Neg) = 99.30%
Therefore, the probability that someone tested negative is approximately 99.30%.
To calculate the probabilities in these questions, we need to use conditional probability and Bayes' theorem. Let's solve each question step-by-step:
1) What is the probability that a person tested positive for Covid19, if they had the virus?
To calculate this, we need to find the probability of a true positive result, given that a person has the virus. From the given information, we know that 30% of tests show a false negative response. Therefore, the probability of a true positive is 1 - 0.3 = 0.7 (because if 30% are false negatives, the remaining 70% must be true positives).
So the answer is 0.7 or 70%.
2) What is the probability that a person who tested negative had Covid19?
To calculate this, we need to find the probability of having the virus, given a negative test result. This can be calculated using Bayes' theorem:
P(Covid19 | negative test) = (P(negative test | Covid19) * P(Covid19)) / P(negative test)
We already know from the given information that 99% of people tested did not have the virus (P(Covid19') = 0.99) and that 30% of tests show a false negative response (P(negative test | Covid19) = 0.3). The probability of a negative test can be calculated by adding the probabilities of true negatives and false negatives:
P(negative test) = P(negative test | Covid19') * P(Covid19') + P(negative test | Covid19) * P(Covid19)
P(negative test) = (1 - P(false negative)) * P(Covid19') + P(negative test | Covid19) * P(Covid19)
P(negative test) = (1 - 0.3) * 0.99 + 0.3 * (1 - 0.99)
P(negative test) = 0.71 + 0.003 = 0.713
Now we can substitute these values into Bayes' theorem:
P(Covid19 | negative test) = (0.3 * 0.01) / 0.713
P(Covid19 | negative test) ≈ 0.0042 or 0.42%
So the probability that a person who tested negative had Covid19 is approximately 0.42%.
3) What is the probability that someone tested negative?
To calculate this, we need to find the probability of a negative test result. We already calculated this in the previous question:
P(negative test) = 0.713 or 71.3%
So the probability that someone tested negative is 71.3%.