Question:
If the false-positive rate of each test in a battery of tests is 0.05, how many independent tests can be included in the battery if we want the probability of obtaining at least one false-positive result to be at most 0.2?

Thank you in advance for the help.

You have an test that produces a false-positive result with probability 0.05, and hence the probability of not obtaining a false-positive is 0.95. Suppose that you perform n, independent repetitions of the test and X is the number of false-positives you obtain. The probability that X is greater than or equal to 1, is 1 minus the probability that X is less than 1. That is

Hence if P(X >= 1) <= 0.2 then 1 - P(X = 0) <= 0.2 and hence P(X = 0) >= 0.8. Thus you want the largest n so that P(X = 0) >= 0.8.

Since the probability of not obtaining a false-positive is 0.95 and the tests are independent, the probability of no false-positives in the n tests is 0.95ⁿ. Thus you want the largest n so that 0.95ⁿ >= 0.8.

Now use your calculator to find 0.95¹, 0.95², ... and see how large the exponent can be before 0.95ⁿ is smaller than 0.8.