Question

There are two towns. In one of them, about 1/6 of them are infected by a virus, while the other is free of disease. You have no idea which town is infected and which is not.

But you do have a test to detect this virus in patients. It is a time consuming test, so you can only do the test twice.

If a person has the virus, then the test shows positive for the virus (with probability 1).

If not, the test still shows a positive with probability 1/5.

You can do the test twice.

Should you administer the test on the same person twice or administer it on two different people? At this point of your knowledge, you will think there a lot of leeway in how you can model this problem. So specify exactly what criteria you use to decide between the two choices (administer on the same person twice or administer on two different people).

Compute all the relevant probabilities and justify your decision based on your criteria.

Answer 1

Given data can be represented as below

p(p/v) = 1 p(n/v) = 0

p(p/nv) = 0.2 p(n/nv) = 0.8

p - denotes the test resulting positive

n - denotes the test resulting negative

v- denotes the perthe son has virus

nv - denotes the person doesnt have virus

Lets find probabilitties for different scenarios

first, only one person who has a virus

the test would result positive both the times. probability that the person has virus = 1*1 = 1

the probability of choosing that person is 1/12

Second, only one person who doesn't have a virus

the probability that test shows positive both the times = 0.2*0.2 = 0.04

the probability of choosing that person is 11/12

if it shows once positive and once negative or both the times negative, it is understood that it is negative.

the probability that the test works = 1*1/12 + (1-0.04)*11/12 = 0.96

Third, one person has a virus and the other didn't

the probability of choosing such pair = 11/144

the probability that test results positive for both = 1*0.2 = 0.2

the probability that test gives exact result = 0.8

total probability that the test works = 0.8*11/144 = 0.06

Therefore, it is better to go with a single peron