Well, first, if 1% of the blood samples really are from people with AIDS,
then you can put 1% of 10,000 = 100 as the margin total in the first column,
and that will leave 10,000 - 100 = 9,900 blood samples that don't have AIDS as
the margin total in the second column. Next, if 99% of the people with AIDS
will be correctly spotted by ELISA, 99 of the 100 samples in the first
column will fall into the first row in that column, leaving 100 - 99 = 1 in
the second row in that column. Next, if 94% of
the 9900 people who don't have AIDS will be correctly told they don't
have it by ELISA, I guess
there should be .94 * 9900 = 9306 blood samples in the second row of the
second column, leaving 9900 - 9307 = 693 for the first row of that column.
Finally, then, the row margin totals are 99 + 594 = 693 and 1 + 9306 = 9307,
the table is complete, and we can work out the conditional probability we
want:
**P(person has AIDS given ELISA positive) = 99/693 = 14%**.
In other words,
only about 14% of the people ELISA says have AIDS actually will have it! This
seems like a disappointingly low figure given ELISA's apparently good
performance numbers (99% and 94%), so it's worth taking a moment to see why.

**The Truth**
person | person does
has AIDS | not have AIDS
------------------------------------------------------
**What ELISA** ELISA positive | 99 | 591 | 693
**says** ELISA negative | 1 | 9,306 | 9,307
-------------------------------------------------------------
100 9,900 | 10,000

There are a variety of points that can be made on the way to an explanation:

Algebraically, a fraction is small when its numerator is small and/or its
denominator is big, and both things are going on here. The numerator (99) is
small because the prevalence of AIDS is low -- since only 1% of the blood
samples will in fact be contaminated, the biggest the numerator could have
been was 100. And the denominator (693) is big because the 594 figure is so
big, which in turn is because the 94% success rate for ELISA among people who
really don't have AIDS isn't so good after all -- 6$\%$ of a big number, 9900
(there's that low prevalence again, indirectly), is a lot of false positives,
as they are called.
The diagonal cells in the 2 by 2 table above are
success stories for ELISA; it's the off-diagonal cells that document the
test's mistakes. Notice the severe imbalance in the two off-diagonal cells --
1 versus 594. ELISA was evidently designed to be terrified of making one of
the two possible mistakes it can make, namely telling people who do have AIDS
that they don't (false negatives, as they're called) -- which makes sense
given that the point is to keep bad blood out of the donated-blood system --
but it is inevitable that in trying hard not to make that kind of mistake
ELISA has to make a fair number of the other kind of error, namely telling
people who don't have AIDS that they do. In practice people who test positive
with ELISA are given a second test (called the Western blot) that's more
expensive but more accurate, and such people are only declared to have AIDS if
both tests come out positive. So, no, the designers of ELISA are not stupid;
*if false negatives are really bad outcomes and the prevalence is low, false
positives are inevitable*.