Question

In the flue season, people with vaccine protection have probability of 0.2 to be infected by the flue virus, whereas people without vaccine protection have probability of 0.6 to be infected by the virus. For a given population, the proportion of people being vaccinated is 0.7.

a.) Given that a person is infected by the flue virus, what is the probability that the person has not taken the vaccine?

b.) Randomly select one person from this population, what is the probability that this person is infected by the flue virus?

Answer 1

We are given here that:  people with vaccine protection have probability of 0.2 to be infected by the flue virus, whereas people without vaccine protection have probability of 0.6 to be infected by the virus

Therefore,
P( infected | vaccinated) = 0.2,
P( infected | not vaccinated) = 0.6

Also, we are given here that: P( vaccinated) = 0.7

a) Using law of total probability, we get here:
P( infected) = P( infected | vaccinated)P(vaccinated) + P( infected | not vaccinated)P(not vaccinated)

P(infected) = 0.2*0.7 + 0.6*(1 - 0.7) = 0.32

Given that the person is  infected by the flue virus, the probability that the person has not taken the vaccine is computed using Bayes theorem as:

P( not vaccinated | infected) = 0.6*0.3 / 0.32 = 0.5625

Therefore 0.5625 is the required conditional probability here.

b) Probability that a person is infected by the flue virus is computed here as:

P( infected) = 0.32 as computed in previous part.

Therefore 0.32 is the required probability here.