yapp - yet another probabilty problem
 

Ok suppose you develop a test - for AIDS say - that is 99% accurate both in ommision and commission [it says yes accurately 99% of the time and no accurately 99% of the time].

Now, you pick a person at random in the UK and give her the test, which turns out positive.

What is the actual chance she has AIDS?

(technically you would probably be testing for HIV rather than AIDS but that's not imprtant to the puzzle - the same sort of principle applies to, say pregnancy tests).

Assume the following approximate figures:
UK population: 60Million
Speed of light: 3x10^8 m/s
AIDS sufferers in UK: 100,000
number of seconds in a year: pi x 10^7


Graeme

99/599, or about 16.5%

It takes awile to get results, so she has extra contraction time for the disease to develop. She may catch aids after, no matter what the test says.

What do the speed of light and the number of seconds in a year have to do with the puzzle?

If the clause "it says yes accurately 99% of the time" means that out of every 100 "yes" results, 99 really have AIDS, then the actual chance that she has AIDS is 99% (and all your other cited figures are irrelevant).

If that clause means something else, then I don't know.

[quote="cheesehead"]If the clause "it says yes accurately 99% of the time" means that out of every 100 "yes" results, 99 really have AIDS.[/quote]

This is a common mistake people make in interpreting these tests for rare conditions.

What the clause means is that 99% of the time the AIDS statius of the person is the same as the AIDS status of the test. If you test 100 people that really have AIDS you will get 99 positive results. If you test 100 people that do not have AIDS you will get 99 negative results. But look what happens when the vast majority of people you test do not have AIDS - those "rare" false positives become more numerous than the "common" true positives.

So the clause "it says yes accurately 99% of the time" does [u]not[/u] mean that out of every 100 "yes" results, 99 really have AIDS.

Instead, it means that out of every 100 real AIDS cases, the test will say yes 99 times (and the clause that it says "no accurately 99% of the time" means that out of every 100 real non-AIDS cases, the test will say no 99 times).

What needed clarification was what the 99% was a percentage of, so that we can properly interpret the statement that the random person picked had a positive result.

The presentation of the problem, then, was tilted toward misinterpretation (or we could say that there was a common mistake in the presentation of the problem). The clause "it says yes accurately 99% of the time and no accurately 99% of the time" implies that one can consider Yes results to be 99% accurate in the sense that out of every 100 Yes results, 99 really have AIDS. That misinterpretation was reinforced by our being told that the person in question had the test "which turns out positive", so that we can consider her to be in the class of people who received yes results.

A less-misleading statement would have been "[99% of AIDS cases are given yes/positive results by the test, and 99% of non-AIDS cases are given no/negative results by the test]".

[quote]What the clause means is that 99% of the time the AIDS statius of the person is the same as the AIDS status of the test.[/quote]
But that still fails to specify what the "99% of the time" applies to: status of the person or status of the test. So it does not really clarify the meaning of the clause.

[quote]If you test 100 people that really have AIDS you will get 99 positive results. If you test 100 people that do not have AIDS you will get 99 negative results.[/quote]
[u]Now[/u] finally there's a real clarification, because it specifies what the 99% was a percentage of. And it turns out that the 99% was [u]not[/u] a percentage of test results, but a percentage of person statuses.

[quote]But look what happens when the vast majority of people you test do not have AIDS - those "rare" false positives become more numerous than the "common" true positives.[/quote]
The way that's written, "false positives" seems to be referring to test results, not persons' status, right? If so, then "rare" is in quotes only because of the apparent conflict in specification of what the rareness applies to. In relation to test results, we know that false positives are only 1%, so are rare without the quotes, right? [u]NO![/u] It turned out above that we don't really know that false positive test results are only 1% of all positive test results! Since we were never informed of that, the adjective "rare" is a figment of imagination, so rightly belongs between quotes.

I still don't see what the speed of light or the number of seconds in a year has to do with this problem.

[quote="Jwb52z"]I still don't see what the speed of light or the number of seconds in a year has to do with this problem.[/quote]

They don't. Haven't you ever taken a test where a problem is over specified to make people use their brains a bit more?

Ok. The probability that she is HIV - positive is
[code:1]
0.1 99
---- x -----
60 100 0.99 0.99
---------------------------------- = ------------ = ------- ~ 1.626 %
0.1 99 59.9 1 0.99 + 59.9 60.89
---- x ------ + ----- x ----
60 100 60 100
[/code:1]

according to [url=http://members.aol.com/johnp71/bayes.html]Bayes[/url]' formula.

Benjamin

I disagree. Given the clarified meaning of the test accuracy, you cannot make any determination.

The number of people in the UK is immaterial.

Consider that she is also in the world population. Should we replace 60 million with 2 billion in the above formula?

Or consider that she is in the population of some village. Should we be using 1 thousand instead.

We need to know about the sample from which the accuracy estimates were drawn. Other populations give us no information about this particular instance.