In a mountain area where mines have to be tested for the emission of pollutants, 32% of all mines emit excessive amounts of pollutants. When tested, 98% of all mines that emit excessive amount of pollutants will fail, but 21% of the mines that do not emit excessive amounts of pollutants will also fail. What is the probability that a mine that fails the test actually emits excessive amounts of pollutants?

Flaws in a carpet tend to occur randomly and independently at a rate of one every 300 square feet. What is the probability that a carpet that is 9 feet by 12 feet contains no flaws?

Say you’re a chemical engineer studying whether a certain trace impurity is present in aproduct. An experiment has a probability of 0.80 of detecting the impurity if it is present. Theprobability of not detecting the impurity if it is absent is 0.90. You have reason to believe that40% of products actually contain the impurity. You perform an experiment which detected animpurity – what is the probability that the impurity is not really there?

A certain disease has an incidence rate of 0.1%. If the false negative rate is 7% and the false positive rate is 3%, compute the probability that a person who tests positive actually has the disease.