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.