A large research hospital has accumulated statistical data on its patients for an extended period of time. Researchers have determined that patients who are smokers have an 18% chance of contracting a serious illness such as heart disease, cancer, or emphysema, whereas there is only a .06 probability that a non smoker will contract a serious illness. From hospital records the researchers know that 23% of all hospital patients are smokers, while 77% are non smokers. For planning purposes, the hospital physician staff would like to know the probability that a given patient is a smoker if the patient has a serious illness.
Prob(disease | smoker) = 0.18
Prob(disease | nonsmoker) = 0.06
In the hospital,
Prob(smoker) = 0.23
Prob(nonsmoker) = 0.77
By definition of Conditional ...
This shows the steps and solution to calculating the probability that a given patient in a hospital (given statistical data) is a smoker if their have a serious illness.