A certain virus infects one in every 2000 people. A test used to detect the virus in a person is positive 96% of the time if the person has the virus and 4% of the time if the person does not have the virus
To determine the probability of someone having the virus given a positive test result, we can use Bayes' theorem.
Let's define the events:
A: Having the virus
B: Testing positive for the virus
We are given:
P(A) = 1/2000 (probability of having the virus)
P(B|A) = 0.96 (probability of testing positive given that the person has the virus)
P(B|~A) = 0.04 (probability of testing positive given that the person does not have the virus)
We want to find:
P(A|B) (probability of having the virus given a positive test result)
Using Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
We can calculate P(B) using the law of total probability:
P(B) = P(B|A) * P(A) + P(B|~A) * P(~A)
To calculate P(~A), we need to find the complement of having the virus:
P(~A) = 1 - P(A)
Now we substitute the given values into the equation:
P(B) = (0.96 * (1/2000)) + (0.04 * (1 - (1/2000)))
Simplifying further:
P(B) = 0.00048 + 0.03996
P(B) ≈ 0.04044
Now we can calculate P(A|B):
P(A|B) = (0.96 * (1/2000)) / 0.04044
Simplifying further:
P(A|B) ≈ 0.023803
Therefore, given a positive test result, the probability of someone having the virus is approximately 0.023803 or 2.38%.