Suppose that 1000 people live in a dorm, and they need to be tested for COVID. Let's assume for simplicity the probability that any given person has the disease is 1/500. Assume that the dorm consists of 10 floors each of 100 people (everyone has a single dorm with no roommates), and that the pipes in the building are designed such that on each floor we have a central hub that lets us access the wastewater from each room, and in the basement, we have a central hub that lets us access the wastewater from each floor. In principle, we can test each room (and thus each of the 1000 people individually), but this process is both expensive and time-consuming. Alternately, we can test the wastewater from each floor first (thus performing only 10 tests initially instead of 1000 ), and if the result for a floor is negative we presume no one on the floor has COVID. If the result for a floor is positive, we now test each of the 100 people on that floor. (In this scenario we suppose that we need to find each individual who is ill, rather than quarantining whole floors of the building.) What is the expected number of tests we have to run?

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