A health authority screens a population for a rare condition…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A health authority screens a population for a rare condition. About people in every have it. The test is positive for of the people who have the condition. It is also positive, by mistake, for of the people who do not have it. A person is chosen at random and tested. The authority wants three numbers. First, the probability that the test is positive. Second, the probability that a person with a positive test has the condition. Third, the probability that a person with a negative test does not have the condition. The authority will use the second number to decide whether a positive result needs a second, more costly test. The third number will tell patients how far a negative result can be trusted. Give the first answer to five places and the others to four places.
Need a reminder? The key ideaShow
A test's sensitivity is , and its specificity is . What a result means depends on the base rate. List the sample space, cross out what the news rules out, and use Bayes' theorem to reverse a condition.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The positive test. Ill and positive: . Healthy and positive: . The sum is .
Step 2. Ill, given positive. .
Step 3. The negative test. Ill and negative: . Healthy and negative: . The sum is .
Step 4. Healthy, given negative. .
Answer: , about and about .
What it means. A positive result is a real warning for only about of the people who get it. A negative result is almost certain to be right.
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