Wednesday, April 01, 2009
What's the Chance That the Patient Has the Disease?
Today, I was reading part of Nassim Nicholas Taleb's book "Fooled by Randomness", and he gave an interesting example of a mistake in calculating probablities:
A test for a disease has 5% false positives (This means that 5% of the tests given to those who do not have the disease come back positive - so if 100 people test positive, only 95 actually have the disease). The disease strikes 1/1,000 of the population. People are tested at random, regardless of whether they are suspected of having the disease. A patient tests positive. What is the probability of the patient having the disease?
Please click here for the answer.
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1 comments:
I guess I have to disagree with the statement that 5% false positives means that "if 100 people test positive only 95 have the disease."..
Actually it means that if 100 people who truly did not have the disease were tested, the test would show that 5% had the disease.
This is how you used it in your solution, so maybe it was just a mistake in the statement of the problem....
Of course in reality, there are false negatives too, (a percentage of the people who actually have the disease will be missed in the testing and given a clean bill of health) and that can add additional complications.
The way I try to get my students to approach this is to make a table and asssume 100 (or 1000000) people, and then divide them up into categories..
Have disease Don't have
Pos test
Neg test
and then put column and row totals to find the numbers... Conditional probability (heck, all probability) is hard for most people, so I want them to break out actual numbers that would apply in some finite population.. it seems to help.
In this situation we want to know how many actually test positive.. and that is all the people who have it, plus 5% of the people who don't. Out of these, only the .1% of the population have it, but 5% of the 99.9% others have it (as your solution shows), so we have a total of about 5% who have tested positive, but only .1% who really are diseased.
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