In a random sample of 1,000 people, it is found that 7.9% have a liver ailment.

Question

In a random sample of 1,000 people, it is found that 7.9% have a liver ailment. Of those who have a liver ailment, 10% are heavy drinkers, 55% are moderate drinkers, and 35% are nondrinkers. Of those who do not have a liver ailment, 10% are heavy drinkers, 48% are moderate drinkers, and 42% are nondrinkers. [a tree diagram may help]. If a person is chosen at random. What is the probability that he or she is a heavy drinker If he or she is a heavy drinker, what is the empirical probability of that person having a liver ailment [Bayes]

Explanation / Answer

Probability that he or she is a heavy drinker = Probability that he has liver ailment and he is a heavy drinker or Probability that he doesnt have liver ailment and he is a heavy drinker =

Probability that he has liver ailment = 0.079 and he is a heavy drinker Probability is .01

Probability that he doesnt have liver ailment = 1- 0.079 = 0.921 and he is a heavy drinker is .01

totyal probability is = 0.079*0.1+0.921*0.1 = 0.1

using bayes theorem given he is aheavy drinker probability that he is having liver ailment is 0.079*.01/(0.079*0.1+0.921*0.1) = 0.0079

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