Question

Although there is not a lot of data available as of yet, scientists approximate that the coronavirus diagnostic test has a probability of 0.95 of giving a positive result when testing a person with the virus and a probability 0.10 of giving a false positive when testing a person without the virus. Assuming 0.01 % of the population are sufferers, calculate the probability that someone does not have the virus, given that they have tested negative.

which one is correct

0.977775

0.988886

0.999994

0.999997

1

Answer 1

We are given here that:
P(+ | disease) = 0.95, therefore P(- | disease) = 1 - 0.95 = 0.05
Also as the probability of false positive is 0.1, therefore,
P( + | no disease) = 0.1, therefore P( - | no disease) = 1 - 0.1 = 0.9

Also, as 0.01 % of the population are sufferers,

Therefore P( disease ) = 0.0001

Using law of total probability, we have here:
P(-) = P(- | disease) P( disease ) + P( - | no disease)P(no disease)
= 0.05*0.0001 + 0.9*(1 - 0.0001)

= 0.899915

Using bayes theorem, we have here:
P( no disease | -) =  P(- | no disease) P( no disease ) / P(-)

= 0.9*(1 - 0.0001) / 0.899915

= 0.999994

Therefore 0.999994 is the required probability here.