A start on the simplest variant of the problem, where we’re assuming the tests are perfect and just trying to minimize the number of tests: the probability of at least one person in the pool having the ’roni in pool size S is going to be 1−(1−P)S, the complement of the probability of them all not having it. We want to choose S such that this quantity is close to 0.5, so S=log0.5log1−P. (For example, if P = 0.05, then S ≈ 13.51.)
My next thought is that when we do get a positive test with this group size, we should keep halving that group to find out which person is positive—but that would only work with certainty if there were exactly one positive (if the “hit” is in one half of the group, you know it’s not in the other), which isn’t necessarily the case (we could have gotten “lucky” and got more than one hit in our group that was chosen to have a fifty-fifty shot of having at least one) …
This reminds me of that cute information-theory puzzle about finding the ball with the different weight! I’m pretty dumb and bad at math, but I think the way this works is that since each test is a yes-or-no question, we can reduce our uncertainty by at most one bit with each test, and, as in Twenty Questions, we want to choose the question such that we get that bit.
A start on the simplest variant of the problem, where we’re assuming the tests are perfect and just trying to minimize the number of tests: the probability of at least one person in the pool having the ’roni in pool size S is going to be 1−(1−P)S, the complement of the probability of them all not having it. We want to choose S such that this quantity is close to 0.5, so S=log0.5log1−P. (For example, if P = 0.05, then S ≈ 13.51.)
My next thought is that when we do get a positive test with this group size, we should keep halving that group to find out which person is positive—but that would only work with certainty if there were exactly one positive (if the “hit” is in one half of the group, you know it’s not in the other), which isn’t necessarily the case (we could have gotten “lucky” and got more than one hit in our group that was chosen to have a fifty-fifty shot of having at least one) …
Partial credit???