Sick or not?
10% of people have a particular disease. A test for the disease gives the correct result 90% of the time (i.e that they do or do not have the disease). If a person is picked at random and the test result is positive, what is the probability that that person has the disease?
Labels: Probability
posted by Chris at
1:24 PM
15 Comments:
Hi Chris, it is 1/2.
It depends on the distribution of the 90% correct test results. If it has the same distribution as the disease itself then the answer is 90%.
But if, say, the distribution is only 72% for those who don't have the disease then that is only 80% of the total population (72% of 90%)then the test would be 100% accurate for those who do have the disease (adding 100% of 10% = 10% to the 80% for those correctly diagnosed as disease-free to get to the overall 90% accuracy).
Lies, damned lies . . . it's wonderful what statistics can be made to do for you!
quantense - you got it.
Wiz. I read it as 90% accurate if have the disease and 90% accurate if don't have the disease. I accept that your point is correct. I always assume that the question contains all you need to know (and then hope it was actually written correctly :))
If the test is correct 90% of the time, and the result is positive, then I would check for the symptoms of the disease. I would look for the symptoms, and if you find them, I would administer the treatment.
You can't just go by percentages alone... 50% of the time their not right. LOL.
OK same situation, but the person tested negative. What is the probability that s/he has the disease?
using a basic statistics tree, you can see that the probability that someone will not only have the disease, but be diagnosed, is 9%. If one in ten people have the disease (10%) and the test accurately diagnoses 90% of cases, multiplying those two probabilities gives you a product of 9%.
Last Anonymous. What is a basic statistics tree?
You have calculated the 9% probability of someone correctly being diagnosed positive if they actually have the disease. But you haven't allowed that the test will also be positive on 10% of the 90% of people who do not have the disease, that is also a 9% probability. So the overall probability of having the disease given only that you tested positive is 9%/(9%+9%)=50%. Thanks for trying though.
Here are two solutions to my problem one post back.
First way: The probability is 1 minus the probability of not having the disease, given that you tested negative.
=> 1-0.9*0.9/(0.9*0.9 + 0.1*0.1)=1-0.81/0.82=0.01/0.82≈1.22%
The second way: 0.1*0.1/(0.9*0.9+0.1*0.1) = 0.01/0.82≈1.22%
I'm surprisingly confident that I've got it right.
Population actually diseased is irrelevant. If test is ninety% accurate and one is diagnosed positive then the test admin. would be ninety% sure that you're sick. right? lol
Hi Lindsay, if the test shows positive for a randomly selected person, then it's 50-50 whether that person is sick or not sick.
Anon here, i may not be reading it right but i am pretty sure the test is irrelavent. if 10% of the population has the disease then the probability of a randomly selected person having the disease is 1 out of ten. or 10%, correct? whether or not the test is correct does not change the fact that the person has or does not have the disease in question.
Hi Anon. What you say is completely correct. Nothing in the solutions contradict what you say (if they did, then the solutions would be in error).
You are considering a different situation to the one(s) that I posted. There we are only considering the probabilities of whether (or not) someone has the disease based on the test result (and it's reliability) only.
To be more specific, if you picked someone who definitely had the disease (I don't know how to do that) and they tested negative, the probability of them having the disease is 100%. What's happening here is that you must have access to a secret test that is 100% reliable, and you are using it's result.
Thank you for asking the question, it helps me to clarify my thoughts. Until the last week or so, I hadn't thought about these things for decades. Greetz.
Also, what hasn't been specified in the original problwm is the nature of the unreliability of the test. Is it a random error - if so, could improve knowledge by repeating the test, or is it because the infected people may not be producing the right kind of anti-bodies (say) that the test depends on. If the latter, repeating the test won't help (probably ;)).
Screw it, euthanize them all!
That's a bit Hitlerian. The disease only causes their toenails to grow half as fast as normal.
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