On Earth 1, there are 10 people with antibodies (red). Nine of them get back correct, positive test results. One person receives a false negative (pale red).
There are also 90 people without antibodies. Eighty-one (90 percent) of them get back correct, negative test results. But 9 people get false positives (pale blue).
9 people correctly test positive (red +).
9 people receive false positives (pale blue +).
That means there is only a 50 percent chance your positive test is one of the real positives. (9/18)
Here, 40 people have antibodies (red). Thirty-six people receive correct, positive test results. But four people receive false negatives (pale red).
There are 60 people without antibodies. Fifty-four people receive correct, negative test results. But six people receive false positives (pale blue).
The FDA estimates the positive predictive value of antibody tests by assuming that 5 percent of the population has antibodies.
Positive predictive values from some tests range from as low as 55 percent to as high as 100 percent (but it's still just an estimate).