Suppose that 1 in 3 people showing symptoms of dry cough, fatigue (tiredness)
and fever have COVID19.There is a test for COVID-19 that gives a positive result 98.5%
of the time when given to someone with the disease. When given to someone without the disease.
80% of the time it gives a negative result.
- What is the probability that a randomly selected person showing symptoms of dry cough, fatigue (tiredness) and fever has COVID-19 if the test returns a positive result?
- What is the probability that a randomly selected person showing symptoms of dry cough, fatigue (tiredness) and fever does not have COVID-19 if the test returns a negative result?
I approached the problem following way but got stuck
Let +ve denote the people who test positive.
And covid denote people who actually have covid.
Now from the given statements we can say that
P(+ve | covid) = .985 — Means Given a person has
covid the probability that the test result will be +ve is .985
P(-ve | non-covid) = .8
P(covid | symptoms) = 1/3
Now the Q1. asks to find out P(+ve | symptoms)
P(+ve | symptoms ) = P(symptoms | +ve) x P(+ve) / P(symptoms)
Now the test gives +ve 98.5% of time for one with covid and
when given to somone without covid it gives +ve 20% of the time.
P(+ve | non-covid) =.2 Dont know how to proceed further
Best Answer
I will answer the first questions, the second is very similiar.
$$p(\text{covid}|\text{positive}) = \frac{p(\text{positive}|\text{covid})p(\text{covid})}{p(\text{positive}|\text{covid})p(\text{covid})+ p(\text{positive}|\text{not covid})p(\text{not covid})} =\\ \frac{0.985\times0.33}{0.985\times0.33 + 0.2\times0.66} \sim 70\% $$
This is why doctors make patients take more tests if a test is known to give false positives. See pic for a geometric interpretation for this kind of problems: think about plugging a number of random event at top.
Bye