Question

Most medical tests used today have about a 5% false positive rate, which some doctors will take to mean that if a patient’s test comes back positive, that the patient has a 95% chance of having the disease. A 95% chance is very high, and so the doctor will assume that the patient has the disease and will start an aggressive and potentially dangerous course of treatment. For your first problem, you will show that, for rare diseases, this common assumption is not even close to accurate.

1. A certain disease is present in 1 out of 1000 people, the probability that a randomly choosen person has this disease is 0.001. A hospital uses a particular test to detect the presence of this disease. If a person has the disease, then the test will detect it 100% of the time, and if a person does not have the disease, the test will detect it 5% of the time. If a patient’s test comes back positive, what is the probability that the patient actually has the disease? Is your answer close to 95% or 0.95? Round your answer to two decimal places. (Proper work includes a tree-diagram for the given scenario and any formulas that are needed.)

One reason for the low probability of a “true” positive result is that the disease is so rare. It is only present in 0.1% of the population, whereas the test will detect it in about 5.1% of the population.

2. Assume that we have the same scenario from Problem 1, except that this time the disease is present in 1 out of 3 people, so that the probability that a randomly choosen person has this disease is 1/3. If a patient’s test comes back positive this time, what is the probability that the patient actually has the disease? Is your answer closer to 95% or 0.95 than last time? Round your answer to two decimal places. (Proper work includes a tree-diagram for the given scenario and any formulas that are needed.)

Answer 1

1) A certain disease is present in 1 out of 1000 people, the probability that a randomly choosen person has this disease is 0.001. The prevelance of the disease is . Define the events

The given conditional probabilities are  . Using total probability theorem,

If a patient’s test comes back positive, the probability that the patient actually has the disease is the conditional probability

It is far lesser than 95%.

2) When . Recalculate all the probabilities.

It is closer to 95%.