In reply to David New.

Thanks for writing. The math does not change if you have false positives (or negatives). The value of p that you use in the formula is the percentage of positive TESTS (not patients). If you have a test that gives many false positives, it will make the pooling process less efficient, as you state.

As you probably know, for any statistical test, there is a trade-off between the proportion of false positives and false negatives. When researchers create the test, they must decide which of the two errors is more acceptable. For a test of a potentially deadly virus, you want the test to not miss any infected people (few false negatives). Consequently, the test will have many false positives. A false positive is very inconvenient (and scary) to the misdiagnosed patient. However, a false negative can lead to dozens of additional cases as people are sickened by the infected individual.