A blog of a writer with high-functioning autistic savant syndrome

Friday, September 15, 2006

Probability Problem

A person is tested for an illness that affects 1 in 10,000. The result is positive, however it's known that the test gives an erroneous positive result in 5% of cases. What is the percentage chance that the person really does have this illness?

Hi there! You are looking at the properties of a screening test result. We have a specificity of 95% ( a test giving a negative result when there truly is no disease) and a 5% false positive rate. You are then asking for the positive predictive value of the test (PPV) when the disease prevalence is 0.0001. Lets assume a 100% sensitivity ( the test returning a positive result when the disease is truly present). Construct a 2*2 table and the PPV can be calculated at 0.2% ie very low because the disease is rare and the specificity 95%. Regards, David (Australia)

Um, is it a trick question? Wouldn't it be a 95% chance as the result is positive, and there is only a 5% error... However I'm pretty bad at maths so I'm getting the feeling I'm completely missing the point. I'll ask my maths teacher tomorrow. :-)

Hang on, I'm not supposed to ignore the first part, am I. Well, does it work out to be 0.000095% chance? Seing as the person would originally have a 0.0001% chance, but you'd have to take 95% of that to allow for the 5% error. Hmm. Still think I'm missing the point...

David's answer is correct. Think of it this way: for every 10,000 people tested around 500 (5% of 10,000) will get a positive result but only 1 of these will actually have the illness. So the odds are about 1 in 500 (0.2%) that your positive result is in fact accurate.

Consider a population of 10000 people. The expected number of Diseased people is 1. The expected number of Undiseased people is 9999. Therefore, the expected number of positive test results is 9999 * 0.05 + 1 = 500.95. Out of these, 1 person actually has the disease. Therefore, the probability of a positive test indicating the disease is 1 out of 500.95 or 1/500.95.

The answer is nearly entirely given by this extract of the problem text: "the test gives an erroneous positive result in 5% of cases". As the test is positive for this person, he has obviously 95% of chances to have this illness.

the exact answer is 20/10019 (near 0.001996). Take the Daniel's idea with 1,000,000 people . 100 will be ill and 999900 will not, so 100 + 5x9999 positive results. the odds of being really sick is 100/(100 + 5x9999).

Hi Daniel. A friend of mine has posted about a briliant minde. About you. You can find it @ http://malibucola.blogspot.com/ it is in Portuguese... but I am 100% sure that you'll get the meaning of his words. You can comunicate in English, as he is from London. You've gain at least 2 "admirors" on the other side of the Atlantic, lol.

Oh yikes! I ignored the 10,000 bit and just went for 95%,as the test has already come up positive and there is a 5% chance that the test is wrong. If he hadn't yet had the test then you guys with clever statistical brains are probably right. (Good job I teach languages, not maths, eh?)

I'm sure the 'wanted' answer is the 1/500 mark (or close to it), and yes, I did too fall for the 95%. But I could argue that the text says "the test gives an erroneous positive result in 5% of cases" and you could wonder 5% of WHICH cases this means. It could mean 5% of the positive cases, in which case the 95% answer is still correct.

And hey, if this interpretation is not possible, I'll just blame it on English not being my native language :)

Would the probability be skewed by the fact that doctors would only test for the disease if the patient displayed symptoms indicating that they might have the disease? Surely in such people the probably that they will have the disease would be greater?

Suppose a person displaying certain correlating symptoms has a 70% chance of having this disease, given the parameters of the a 5% false positive rate and disease prevalence of 0.00001. Is it ever worth doing the test? Secondly, if the symptoms and the test are the only indications of the disease, how could they ever come to that 70% figure?

## 16 comments:

Hi there!

You are looking at the properties of a screening test result. We have a specificity of 95% ( a test giving a negative result when there truly is no disease) and a 5% false positive rate. You are then asking for the positive predictive value of the test (PPV) when the disease prevalence is 0.0001. Lets assume a 100% sensitivity ( the test returning a positive result when the disease is truly present). Construct a 2*2 table and the PPV can be calculated at 0.2% ie very low because the disease is rare and the specificity 95%. Regards, David (Australia)

Um, is it a trick question? Wouldn't it be a 95% chance as the result is positive, and there is only a 5% error... However I'm pretty bad at maths so I'm getting the feeling I'm completely missing the point. I'll ask my maths teacher tomorrow. :-)

Hang on, I'm not supposed to ignore the first part, am I. Well, does it work out to be 0.000095% chance? Seing as the person would originally have a 0.0001% chance, but you'd have to take 95% of that to allow for the 5% error. Hmm. Still think I'm missing the point...

My statistics teacher is screaming Bayes Rule somewhere...

David's answer is correct. Think of it this way: for every 10,000 people tested around 500 (5% of 10,000) will get a positive result but only 1 of these will actually have the illness. So the odds are about 1 in 500 (0.2%) that your positive result is in fact accurate.

Yep 1 in 500. straight up yr 10 stats/probability problem.

Consider a population of 10000 people. The expected number of Diseased people is 1. The expected number of Undiseased people is 9999. Therefore, the expected number of positive test results is 9999 * 0.05 + 1 = 500.95. Out of these, 1 person actually has the disease. Therefore, the probability of a positive test indicating the disease is 1 out of 500.95 or 1/500.95.

the result is one chance over 500

The answer is nearly entirely given by this extract of the problem text: "the test gives an erroneous positive result in 5% of cases". As the test is positive for this person, he has obviously 95% of chances to have this illness.

the exact answer is 20/10019 (near 0.001996). Take the Daniel's idea with 1,000,000 people . 100 will be ill and 999900 will not, so 100 + 5x9999 positive results.

the odds of being really sick is 100/(100 + 5x9999).

Hi Daniel. A friend of mine has posted about a briliant minde. About you. You can find it @

http://malibucola.blogspot.com/

it is in Portuguese... but I am 100% sure that you'll get the meaning of his words. You can comunicate in English, as he is from London.

You've gain at least 2 "admirors" on the other side of the Atlantic, lol.

Take care and all the best.

Carlos Roque.

Oh yikes! I ignored the 10,000 bit and just went for 95%,as the test has already come up positive and there is a 5% chance that the test is wrong. If he hadn't yet had the test then you guys with clever statistical brains are probably right. (Good job I teach languages, not maths, eh?)

I'm sure the 'wanted' answer is the 1/500 mark (or close to it), and yes, I did too fall for the 95%. But I could argue that the text says "the test gives an erroneous positive result in 5% of cases" and you could wonder 5% of WHICH cases this means. It could mean 5% of the positive cases, in which case the 95% answer is still correct.

And hey, if this interpretation is not possible, I'll just blame it on English not being my native language :)

Would the probability be skewed by the fact that doctors would only test for the disease if the patient displayed symptoms indicating that they might have the disease? Surely in such people the probably that they will have the disease would be greater?

-antipodeanglassgirl

Suppose a person displaying certain correlating symptoms has a 70% chance of having this disease, given the parameters of the a 5% false positive rate and disease prevalence of 0.00001. Is it ever worth doing the test? Secondly, if the symptoms and the test are the only indications of the disease, how could they ever come to that 70% figure?

Hi Daniel.

Check out Jeffrey Rosenthal's site struckbylightning.ca He has all sorts of crazy fun with probability.

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