An AIDS test shows the right result 99% of the time.

efarmer@math.uiuc.edu wrote:

> The problem, IMO, is interpreting what we mean when we say that the
> test is correct 99% of the time.

I agree.

> Imagine a group of people NONE of whom have AIDS.  99% of them will
> test negative (i.e., the test will report CORRECTLY that they do not
> have AIDS).  Imagine a group of people ALL of whom have AIDS.  99% of
> them will test positive.  In other words, given that you have AIDS
> (resp. don't have AIDS), there is a 99% chance that you will test
> positive (resp. negative).
>
> BUT the question asked the "reverse" conditional probability; GIVEN
> that you test positive, what are the chances you actually have AIDS?
> Symbolically, let A be "I have AIDS", and T be "I test positive."
> Then we know three probabilities: P(A) = 0.000001, P(T | A) = 0.99,
> and P(~T | ~A) = 0.99.  The probability we are asked to calculate is
> P(A | T).
>
> Eric Farmer

That is almost certainly the intended question.

But here is the actual question:

>An AIDS test shows the right result 99% of the time. Say, one in a
>million people have AIDS in whatever area we are. We go and take the
>test and oh no we test positive for AIDS. What are the chances we
>actually have AIDS?

That is ambiguous.

There are many ways for the test to show "the right result 99% of the
time".  For example, given that really only one in a million people have
AIDS, I could devise a test that gives the right result 99% of the time
by using a random number generator.  If I just have it say POSITIVE!
PATIENT HAS AIDS! 1% of the time, and NEGATIVE! PATIENT DOES NOT HAVE
AIDS! 99% of the time, my test will be right almost exactly 99% of the
time.  I could adjust my frequencies a little, while still making my
test utterly insensitive to the presence of any virus, and get an
exactly 99% 'success rate'.

If that were my test, obviously the chance of having AIDS given a
positive test result would be precisely one in a million.

(Roughly a year ago a poster to rec.puzzles pointed out this problem
with the most common ways of stating these 'base rate fallacy' puzzles.
I'm afraid I can't remember his name, he was a doctor, I think, living
in New Zealand.)

-- Jamie Dreier <pl436000@brownvm.brown.edu>

[ Editor: the numerical value of P(A | T) under the intended reading is
  not worked out in the collected postings. ]