# Disease

Diapason
Posers and Puzzles 23 Dec '06 10:23
1. 23 Dec '06 10:23
There are exactly 60 million people in the UK. Of these, exactly 6000 are infected with a new disease. The only test available that can tell whether someone has the disease gives the correct reading 99% of the time (on average).
You are tested and - horrors! - the test is positive.
What is the probability that you are actually infected?
2. 23 Dec '06 11:20
I hopeI didn't miscalculate: 0.98% ?
3. 23 Dec '06 12:27
There are 4 cases:
(A) You have the desease and the test say you have.
(B) You don't have the desease and the test says you haven't.
(C) You have the desease but the test says you haven't.
(D) You don't have the desease but the test says you have.

Now, (A) plus (B) adds up to 99% of the test cases.
And (C) plus (D) adds up to 1% of the test cases.

If the test says you have the desease only the cases (A) and (D) is interesting. But (A) os only a part of 99% and (D) is only a part of 1%, but how large these parts are we cannot say.

So, in my opinion, the exact probability that you have the desease will stay unknown. But one thing I know is that the precision of the test is a lot more if you take the test again and if the second test says the same thing, then you can be pretty sure that you have the desease.
4. 23 Dec '06 12:321 edit
my answer was not a joke, you know ... Just use Bayes' Theorem.

P(A|B) = P(A) x P(B|A) / P(B)
5. 23 Dec '06 13:16
if its correct 99 times out of 100, then doesnt that mean you have a 1% chance of not having the disease?
6. 23 Dec '06 13:54
No, because there are many more 'positive' test results from the un-infected population ( 1% of 5 994 000) than from the infected ones (99% of 6000).
7. 23 Dec '06 14:11
Originally posted by Mephisto2
No, because there are many more 'positive' test results from the un-infected population ( 1% of) than from the infected ones (99% of 6000).
So if you get a positive result ...
[I don't think this agrees with your earlier answer - I was thinking along the lines of this one. What do you think?]
8. 23 Dec '06 15:03
Originally posted by Diapason
So if you get a positive result ...
[I don't think this agrees with your earlier answer - I was thinking along the lines of this one. What do you think?]
Right, I was afraid of mis-calculation. It should have produced the same result, though. How about 9% approx?
9. 23 Dec '06 15:37
Originally posted by Mephisto2
Right, I was afraid of mis-calculation. It should have produced the same result, though. How about 9% approx?
That's what I get.

I like this. If a particular test with an accuracy of 99% gives you the thumbs down then you only have a 9% chance of having the disease. Nicely counter-intuitive!
10. 23 Dec '06 15:56
The probability that I am infected is 99%. The other numbers in the problem are irrelevant. If the test is accurate 99% of the time, then there is a 1% chance that it is wrong with respect to me, i.e., that I do not have the disease. There is a 99% chance that my results are correct, and that I do have the disease. I sure hope it's not terminal.
11. 23 Dec '06 16:35
Originally posted by hahn121
The probability that I am infected is 99%. The other numbers in the problem are irrelevant. If the test is accurate 99% of the time, then there is a 1% chance that it is wrong with respect to me, i.e., that I do not have the disease. There is a 99% chance that my results are correct, and that I do have the disease. I sure hope it's not terminal.
"The probability that I am infected is 99%"

That is the wrong conclusion. As said before, would you let the 59940 people (1% of 599 400) people who are not infected but tested positively, let believe they are infected?
12. BigDoggProblem
Just...the Dogg
23 Dec '06 18:551 edit
Hmm, not sure where 9% is coming from.

605 880 people will test positive. Of these, only 5 940 are actually infected. How is the answer not 0.98%?
13. 23 Dec '06 19:36
Originally posted by Diapason
There are exactly 60 million people in the UK. Of these, exactly 6000 are infected with a new disease. The only test available that can tell whether someone has the disease gives the correct reading 99% of the time (on average).
You are tested and - horrors! - the test is positive.
What is the probability that you are actually infected?
There aren't exactly 60 million people in the UK. Flawed before you begin any conjecture!
14. BigDoggProblem
Just...the Dogg
23 Dec '06 19:40
Originally posted by mikelom
There aren't exactly 60 million people in the UK. Flawed before you begin any conjecture!
It's a math problem. The population figures, disease, etc. are obviously fictional. ðŸ™„
15. 23 Dec '06 19:42
Originally posted by BigDoggProblem
It's a math problem. The population figures, disease, etc. are obviously fictional. ðŸ™„
Apologies, I wasn't being facetious. May be now I have a disease I never knew I had.