A quiz: math problem/playing with numbers

T

tj

New Member
I found the following interesting. I'll post the answer later. In the meantime, if you're looking for a challenge, try to figure out the following (warning, it probably seems pretty hard at first).

A quiz given to medical doctors, borrowed from the book “Randomness” by Deborah Bennett:

 A test of a disease presents a rate of 5% false positives. The disease strikes 1/1000 of the population. People are tested at random, regardless of whether they are suspected of having the disease. A patient’s test is positive. What is the probability of the patient having the disease?

(The original author states that one in five doctors got this question correct.)
 
Peaches said:
Heh. I remember this exact same problem from Day 1 of Prob & Stats in high school...
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Yeah, I think what throws me off in the question is how the test is 95% accurate, but the actual odds of you having the disease (when you test positive during the random testing) is much much lower.
 
Gonna hafta work this out, it's driving me nuts. Unfortunately, it's been years since I've looked at anything resembling probability, so this is going to take an embarassingly long time. *sigh*
 
Lol, this was only posted for entertainment value, so if it's actual hard work/no fun, I can just PM you the answer. (I'm planning on posting the answer probably tomorrow).
 
No, I love puzzles. They'll just drive me nuts until I figure them out. Believe me, it's entertainment.
 
Either its a) a red herring then the chances of having the disease is still 1/1000
or b) the chances of it being a false positive are 5/100 and for every 1000 people tested
there will be 51 positive results then its a 1/51 chance of being a true positive.
or c) i don't feel very well.
 
The question could do with some clarifications - esp. the definition of a 'false positive'.

Assuming 'false positive' as meaning "a healthy individual labeled as diseased", the answer is 0.95.
Once the test comes out +ve, the 1/1000 probability has no bearing on the problem and the individual has the disease if the test is right, and the individual definitely doesn't have the disease if the test is wrong.

If the definition of 'false positive' is "the test just says that the disease is present in an individual but the result has no correlation with the underlying fact of whether or not the individual has the said disease", the probability is (0.95 * 1 + 0.05*0.001) = 0.95005.
In this case, if the test is true, then the probability is 0.95. But if the test is false, then it says nothing about whether or not the patient has the disease. So then we have to fall back on the probability of 1/1000 that applies to all individuals in the pouplation to compute the probability of the test being a false positive and the patient still having the disease anyway.
 
tj said:
A test of a disease presents a rate of 5% false positives.
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What does that mean? Does it mean that 5% of all positive test results are incorrect or that 5% of all test results are incorrectly positive?
 
DrDoug said:
What does that mean? Does it mean that 5% of all positive test results are incorrect or that 5% of all test results are incorrectly positive?
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common scientific usage of the phrase 'false positive' means the former (i.e. if a test is +ve, then the result is 95% reliable because out of every 100 +ve results, 5 are incorrect)
 
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