yapp - yet another probabilty problem

[toferc]

Originally Posted by Wackerbarth

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

The number of people in the UK is immaterial.

The statement of the problem was

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

The population of the UK, along with the number of AIDS sufferers in the UK, is exactly the information you need to answer the question.

[smh]

Originally Posted by S80780

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]

Isn't this more like the chance she does not have AIDS?

This way the test would be very unreliable.

[toferc]

Originally Posted by S80780

Ok. The probability that she is HIV - positive is
<snip>
Benjamin

You misplaced a decimal. The calculation should read
[code:1]
0.1 99
---- x -----
60 100 9.9 9.9
---------------------------------- = ------------ = ------- ~ 14.18 %
0.1 99 59.9 1 9.9 + 59.9 69.8
---- x ------ + ----- x ----
60 100 60 100
[/code:1]

[toferc]

Originally Posted by smh

Isn't this more like the chance she does not have AIDS?

This way the test would be very unreliable.

The whole point of exercises like this is that when testing for a rare condition, false positives can (and often do) outnumber true positives, even when the accuracy on a single test is high. The test described is correct 99% of the time, but in this case over 85% of the positives would be false.

[graeme]

Oh good lots of replies to this and the drug problem I posted. The correct answer has been posted but I'll go through it step by step anyway.

If you were to test the entire population of the UK with this test, you would get
1/100 * ~60M = 600,000 false positives
plus
99/100 *100,000 = ~100,000 true positives

so you get 700,000 positives of which 100k were true or 1/7 true positives.

Therefore if you randomly choose a person and they test positive - in the 99% accurate test - they only have a 1in 7 chance of actually having AIDS (or 14% or 86% of not having it, etc).

As has been pointed out, this extremely counter intuitive result is the bugbear of all tests of this type, particularly mass screening where the sheer numbers in the poulation make false positives a significant amount of the total positives. Do you make the test give less false postives - typically missing more true positives, or do you catch all the real positives, but risk getting drowned in all the false positives? This is made particularly hard when the test could mean life or death for the subjects.

In the case of AIDS tests, older readers may remember stories of people commiting suicide becasue they had been told they had it, only for their families to be told thast it was a false positive.

Also in the real world, people tend to take AIDS tests if they feel that they may have it, or are at risk of having it, rather than being randomly selected (similarly for pregnacny tests). This will skew the results accordingly. But in the case of say breast cancer screening, where every woman is expected to be tested at some point (in the UK anyway), this sort of problem is a real concern to the test administrators.

Graeme