First, it assumes that a prior false positive does not increase the likelihood of a false positive on testing again (that is, a retest's result is independent f

First, it assumes that a prior false positive does not increase the likelihood of a false positive on testing again (that is, a retest's result is independent from the first test's result). Given that, from the first test, we have 100 people who were false positives, and 1 person who is truly positive. We would expect that 1 truly positive person to be positive again (we have a 99% chance of it from the test, after all), and 1 of the 100 who were false positive before to also test positive this second time (that 1% inaccuracy rate). Given 1 true positive, and 1 false positive, in that sample, you would expect a positive rate to mean you have a 50% chance of being the true positive. If we did not have information from the first test, yes, it would be ~1% chance of a false positive. We have nearly a 0% chance of having a true positive, so of the 101 people, and a 99% accurate test, we would expect one to come out positive, despite not having the disease. But, because we have that first test, from a much larger sample size, we likely have one person who is a true positive, and so could expect two people to test positive, one with the disease, and one without (the 1/100 of the remaining), giving us a 50% chance a positive result = has the disease.