Question 1114805: There are 100 patients in a hospital with a certain disease. Of these, 10 are selected to undergo a drug treatment that increases the percentage cured rate from 50 percent to 75 percent. What is the probability that the patient received a drug treatment if the patient is known to be cured?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! this is what i think it will be.
the probability that a patient receives the drug treatment is 10/100 = .1
the formula for that is p(D) = .1
10% of the time, the patient is cured using the drug, and 90% of the time the patient is cured without taking the drug.
the formula for that is p(C) = .1 * .75 + .9 * .5 = .525
the probability that a person is cured and has taken the drug is .1 * .75 = .075
the formula for that is p(C and D) = .075
given that a person is cured, what is the probability that the person took the drug.
the formula for that is p(D given C) = p(C and D) / p(C).
that becomes p(D given C) = .075 / .525 = .1428571429.
i usually try an experiment to see if what i did is reasonable.
my experiment for this problem is as follows.
assume 1000 patients.
10% of them receive the drug and 90% of them don't.
that makes 100 who received the drug and 900 who didn't.
75% of the ones who took the drug are cured = 75
50% of the ones who didn't take the drug are cured = 450.
total number of cured patients = 525.
525/1000 = .525, so p(C) looks like an accurate ratio.
given that a person is cured, what is the probability that the person took the drug.
525 were cured and, out of them, 75 took the drug.
75/525 = .1428571429, so p(D given C) looks good as well.
75 patients out of the 1000 gook the drug and were cured.
p(D and C) = 75 / 1000 = .075, so that ratio looks good as well.
the solution looks reasonable, so i would go with that.
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