Question

Last year medical students are sent to rural places in order to
medicate and relieve the inhabitants of this population who do not have access to health
quality. An aspiring doctor finds that a quarter of a population
is vaccinated against malaria. In an epidemic of this disease, he observes that
of every 5 patients 1 is vaccinated. It is also known that of every 12 vaccinated only
1 is sick. The doctor wants to calculate the probability that a non-vaccinated person is
sick

Answer 1

Also, we are given that a quarter of a population is vaccinated against malaria, therefore
P( vaccinated ) = 0.25, therefore P(non vaccinated ) = 1 - 0.25 = 0.75

We are given here that: of every 5 patients with disease, 1 is vaccinated, and of every 12 vaccinated only
1 is sick, therefore we get here:
P( vaccinated | sick ) = 1/5 = 0.2
P( sick | vaccinated ) = 1/12

Using bayes theorem, we have here:
P( sick and vaccinated ) = P( sick | vaccinated ) P(vaccinated ) = (1/12)*0.25 = 1/48

Using bayes theorem, again, we have:
P( sick ) = P( sick and vaccinated ) / P( vaccinated | sick ) = (1/48) / 0.2 = 5/48

Using law of total probability, we have:
P(sick) = P( sick and vaccinated ) + P(sick | non vaccinated ) P(non vaccinated )
(5/48) = (1/48) + 0.75 P(sick | non vaccinated )
Therefore, P(sick | non vaccinated ) = (1/12)/0.75 = (1/12)*(4/3) = 1/9

Therefore 1/9 = 0.1111 is the required probability here.