# You Test Positive

davegage
Posers and Puzzles 09 May '05 04:49
1. 09 May '05 04:49
Suppose you go to the doctor to be tested for a rare disease known to inflict only about 1 in every 10,000 people.

Suppose the test results are 99% accurate (this applies to both a positive and a negative diagnosis).

To your dismay, the test comes back positive. How worried should you be?
2. AThousandYoung
09 May '05 05:21
This one's got me stumped š.
3. Alcra
Lazy Sod
09 May '05 05:53
Originally posted by davegage
Suppose you go to the doctor to be tested for a rare disease known to inflict only about 1 in every 10,000 people.

Suppose the test results are 99% accurate (this applies to both a positive and a negative diagnosis).

To your dismay, the test comes back positive. How worried should you be?
Let me begin by saying that I am a statistical dolt!

But, the chance of the test being wrong is greater than my having the diseased (1/100 as oppossed to 1/10000).

Therefore, I would be VERY worried indeed, but needlessly!

4. 09 May '05 07:50
1/101
5. Daemon Sin
I'm A Mighty Pirateā¢
09 May '05 15:02
Not that 'm any good at maths but 99% accuracy at 1/10000 chance and it's positive.....

Isn't that like 99.88888888888888888888888888888888888888889% of actually having the disease?!

I'd have given up hope!
6. 09 May '05 15:13
Well, If the results are 99% accurate, then there is only 1% chance you are not sick.

So I would be worried.
7. 09 May '05 15:132 edits
Originally posted by Daemon Sin
Not that 'm any good at maths but 99% accuracy at 1/10000 chance and it's positive.....

Isn't that like 99.88888888888888888888888888888888888888889% of actually having the disease?!

I'd have given up hope!
Ask Katty Queen. She should be able now to work out the Bayes' Theorem application to find out why it is 1/101, or slightly less than 1%. Which sounds surprising, but with an error margin that is 100 times larger than the expectation, the test is not that effective.

edit: just to get a feeling: on a population of 10000, only 1 case is expected. The test has 99% accuracy, and would give about 100 positive results on people not infected (1% ). So, the 1 real case vs. the 100 test misses ....
8. 09 May '05 18:37
Originally posted by Mephisto2
edit: just to get a feeling: on a population of 10000, only 1 case is expected. The test has 99% accuracy, and would give about 100 positive results on people not infected (1% ). So, the 1 real case vs. the 100 test misses ....
Yes, this is the idea.

The problem with the test is that the 99% accuracy applies to both positive and negative diagnosis. You are much more likely to get diagnosed positive without actually having the disease than you are to get diagnosed positive with actually having the disease.

A follow up question: Suppose you ask for a retest, and it's positive again. How worried are you now?
9. Alcra
Lazy Sod
10 May '05 10:47
Originally posted by davegage
Yes, this is the idea.

The problem with the test is that the 99% accuracy applies to both positive and negative diagnosis. You are much more likely to get diagnosed positive without actually having the disease than you are to get diagnosed positive with actually having the disease.

[b]A follow up question: Suppose you ask for a retest, and it's positive again. How worried are you now?
[/b]
The second question is answered the same way - no need to be worried.

Repeats of an experiment are not influenced by prior results. Therefore, the odds are the same the second time around as the first.
10. 10 May '05 11:374 edits
Originally posted by Alcra
The second question is answered the same way - no need to be worried.

Repeats of an experiment are not influenced by prior results. Therefore, the odds are the same the second time around as the first.
I doubt it. I would start getting worried with 99/199 chance which is almost 50%.

edit. and it'd become really serious if a second retest were positive, that would make it 98.99% or 99% probable of being infected.
11. Palynka
Upward Spiral
10 May '05 14:45
Originally posted by Mephisto2
I doubt it. I would start getting worried with 99/199 chance which is almost 50%.

edit. and it'd become really serious if a second retest were positive, that would make it 98.99% or 99% probable of being infected.

The probability of a double test being wrong is 1%*1%, that means that the test will fail in average once in 10.000 tries.

So in 10000 tests in average there will be two positive tests (ok, a little lower as there is a small chance of the test being negative exactly when it should be positive).

Doesn't that mean that the probability of a person with both tests positive actually being infected should be around 50%?
12. 12 May '05 07:44
Originally posted by Palynka