It's Been A While Since We Had A Brain Teaser.

[JT!]

Lets say that you live on an island.

On this island there are 100,000 people.

And on this Island 1 out of every 1000 people are HIV positive.

The test to see if you are HIV positive is 99% accurate. Meaning if a person isn't HIV positive, 1% of the time the test will say they are, and if a person is HIV positive, 1% of the time the test will tell them they aren't.

If you get your test result and it says you are HIV positive according to the test results, what are the chances that you are indeed HIV positive?

Edited July 2, 2011 by JT!

[Fish-Finger-er]

the plane takes off, definitely.

[chris4stars]

The test to see if you are HIV positive is 99% accurate:

For the 100 people who are positive, 99 of them will receive the accurate positive result (on average)

Meaning if a person isn't HIV positive, 1% of the time the test will say they are

For the 100,000 - 100 = 99,900 people who arent HIV positive, 999 of them will receieve an inaccurate positive result

Total people recieving the positive result on the island = 99 + 999 = 1098

Total people who actually are postive on the island = 100

Chances that positive result actually constitutes as an actual HIV positive = 100/1098 = 0.091....etc

that amounts to 9.1% rounded up?!

The amount of accurate results the test itself gives = 99 + (99,000 - 999) = 98100 out of a possible 100,000 meaning the overall test accuracy is actually only 98.1% accurate?

All guess work here...no idea if the logic is right

Edited July 3, 2011 by chris4stars

[Agram]

interesting Q... but i suck at math... so im gonna wait to c what happens

in the first place.. if i knew the statisctics about that island.. id move off of it!

But respect for the elaborat Q u really put some time into it

Edited July 2, 2011 by Agram

[JT!]

I'll let it stew and see what people come up with, but the idea of the question is to make you realize something that seems very accurate (99% is the answer most people estimate) isn't actually right when you think it over some more.

[Agram]

I'll let it stew and see what people come up with, but the idea of the question is to make you realize something that seems very accurate (99% is the answer most people estimate) isn't actually right when you think it over some more.

now that u simplyfied it I can apsolutley agree with that statement..

yeah..let it stew for a while.. im interested what answers u gonna get

Edited July 2, 2011 by Agram

[chris4stars]

Argh...i messed up on the basics in mine initially! 1% of 99900 = 99.9?! im claiming its the beers last night fault! its been edited...curious if its close

[PaRtZ]

I think chris4stars has it. Can't fault the steps and logic seems right. Although 9.81% seems a little too high a number for the 'OMGZ' effect I think JT is after....

[JT!]

Chris is right. Believe it or not, in this situation, the chances of you being HIV positive when your test result is posative is just less than 1 in 10, even though the test is considered 99% accurate, if your test results come back positive, there's a huge chance that you aren't actually HIV positive. (Obviously if you're up on your probability and don't assume that with a positive test result means you're 99% likely to actually be HIV positive it'll have a much lesser impact on you and this might not be very impressive).

To step away from the maths, looking at it logically the chances of you having HIV is 1 in 1000. But the test inaccuracy is 1 in 100. The chances of the test been wrong is 10 times more likely than if you actually have HIV in the first place which is where the (about) 10% comes from.

Edited July 3, 2011 by JT!

[RobinJI]

Surely you then just take the test again, then if your individual test came back positive, there would only be a 1% chance that it was wrong, and only a 0.0001% that both consecutive tests had been wrong?

But yeah, when the sampling group is large 99% starts not actually being a great success rate. Imagine if 99% of cars left the show rooms with working brakes. The other 1 in 100 not having working brakes wouldn't be even close to acceptable.

[arw_86]

is it me or is the answer just 1%?

.....it's just me!

Edited July 3, 2011 by arw_86

[Greetings]

edit: never mind

Edited July 3, 2011 by Greetings

[chris4stars]

Surely you then just take the test again, then if your individual test came back positive, there would only be a 1% chance that it was wrong, and only a 0.0001% that both consecutive tests had been wrong?

But yeah, when the sampling group is large 99% starts not actually being a great success rate. Imagine if 99% of cars left the show rooms with working brakes. The other 1 in 100 not having working brakes wouldn't be even close to acceptable.

I dont think this is quite true:

remember, if the test comes back initially as positive, there is a 9.1% chance that you ARE actually HIV +

...anyway, we now forget about the initial test and only deal with these 1098 (unfortunate people), the initial test result has no real effect on the second as the test remains the same and doesnt have a magical memory...its completely separate

In this new scenario, 1098 people came back with a positive result...if they were to retake the test (its naturally only them who would be inclined to) we would be presented with this:

1098 on the island of "1st positive test results" (for want of a better name), but only 9.1% of the population have it (100 - as mentioned originally)...the rest of the maths is exactly the same as the first question

The test to see if you are HIV positive is 99% accurate:

For the 100 people who are positive, 99 of them will receive the accurate positive result (on average)

For the 998 who are actually negative, the 1% inaccuracy still gives 9.8 people recieving a false positive result (giving them two positives in a row)

TOTAL PEOPLE recieving double positive result = 109 (rounded up the 9.8)

TOTAL PEOPLE who are actually positive = 100

chances that the second positive result = an ACTUAL positive = 100/109 = 92% (ish)

Edited July 3, 2011 by chris4stars

[Tomm]

This is exactly the reason that as a doctor, you shouldn't request tests for people who are not unwell. Essentially you've demonstrated that pre-test probability has a huge effect on the predictive value of the test. Which is why tests that work in a hospital setting will often be useless in screening the general population. It's also why having a full body CT scan for no particular reason (as many private companies will offer in the US) is a stupid thing to do - consider the risk of 'false positive' results and the unnecessary investigations that might result.

Chris4stars has it right. You can demonstrate this by drawing a 2x2 table: Totals add up to 100,000.

				Disease

			-ve		+ve	

Totals			99 900		100



				Disease

			-ve		+ve	


Test +ve		999		99           


Test -ve		98901		1

Take the horizontal line for 'test +ve' and you can see that there far more people in the 'disease -ve' column. This is a complicated problem - even more so when you consider that your 'false negative' and 'false positive' rates are often wildly different (in your example, both are 1%). Tests are very rarely 100% positive or negative - most biological parameters are continuous variables and you choose a threshold, below or above which a test is considered to be positive or negative. Depending on the context, you might want a test with fewer false positives, and you would have to compromise by allowing more false negatives.

Surely you then just take the test again, then if your individual test came back positive, there would only be a 1% chance that it was wrong, and only a 0.0001% that both consecutive tests had been wrong?

In theory, you're right. However, you have to consider that in a biological system, two test results from the same patient are likely to be the same - if your test was a 'false' positive, it's likely because of a 'patient factor' (strange antigens etc) rather than simple probability.