A certain virus infects one in every 300 people. A test used to detect the virus in a person is positive 80% of the time if the person has the virus and 5% of the time if the person does not have the virus. (This 5% result is called a false positive.) Let A be the event "the person is infected" and B be the event "the person tests positive".
Hint: Make a Tree Diagram
a) Find the probability that a person has the virus given that they have tested positive, i.e. find P(A|B). Round your answer to the nearest tenth of a percent and do not include a percent sign.
P(A|B)= %
b) Find the probability that a person does not have the virus given that they test negative, i.e. find P(A'|B'). Round your answer to the nearest tenth of a percent and do not include a percent sign.
P(A'|B') =
A. 3.34
B. 99.9
See similar problem here:
http://www.jiskha.com/display.cgi?id=1481848396
To solve these probability questions, we can use Bayes' theorem.
a) To find P(A|B), the probability that a person has the virus given that they have tested positive, we can apply Bayes' theorem.
P(A|B) = P(B|A) * P(A) / P(B)
1. P(B|A) is the probability of testing positive given that the person has the virus. In this case, it is given as 80%, or 0.8.
2. P(A) is the probability of a person being infected, which is given as one in every 300, or 1/300.
3. P(B) is the probability of testing positive. To find this, we need to consider both cases: a true positive (A and B) and a false positive (A' and B).
P(B) = P(A and B) + P(A' and B)
P(A and B) = P(B|A) * P(A) = 0.8 * 1/300
P(A' and B) can be calculated as:
P(A' and B) = P(B|A') * P(A') = (1 - P(B|A')) * P(A')
= (1 - 0.05) * (299/300)
Then, we can calculate P(B):
P(B) = P(A and B) + P(A' and B)
Now, we can substitute these values into Bayes' theorem:
P(A|B) = P(B|A) * P(A) / P(B)
After plugging in the known values, we can solve for P(A|B).
b) To find P(A'|B'), the probability that a person does not have the virus given that they test negative, we can again use Bayes' theorem.
P(A'|B') = P(B'|A') * P(A') / P(B')
1. P(B'|A') is the probability of testing negative given that the person does not have the virus, which is given as (1 - 0.05) or 0.95.
2. P(A') is the probability of a person not being infected, which is 299/300.
3. P(B') is the probability of testing negative. To find this, we need to consider both cases: a true negative (A' and B') and a false negative (A and B').
P(B') = P(A' and B') + P(A and B')
P(A' and B') can be calculated as:
P(A' and B') = P(B'|A') * P(A') = 0.95 * (299/300)
P(A and B') can be calculated as:
P(A and B') = P(B'|A) * P(A) = (1 - P(B|A)) * (1/300)
Now, we can calculate P(B'):
P(B') = P(A' and B') + P(A and B')
Finally, substitute the known values into Bayes' theorem:
P(A'|B') = P(B'|A') * P(A') / P(B')
After solving, round your answers to the nearest tenth of a percent and do not include the percent sign.