Poisson Distribution

Yes: 20 cups, each containing 1cc of water, exactly one of which also contains a coliform bacterium.

Explanation

The rate r in our original problem was 1 bacterium per 20cc of water. We have here divided that 20cc up into 20 units of 1cc each, and specified that the group of 20 must contain exactly 1 bacterium.

It should be obvious that this situation is no longer based on rate of occurrence (which can vary); the proportion of bacteria in our 20-cup sample is precisely fixed. We can ask which cup has the bacterium, but we cannot any longer ask whether our 20cc sample will have more or less than 1 bacterium. If it is not free to vary, it is no longer a rate, it is a proportion. With a 20cc sample of water taken from the whole river, there is a chance of finding no coliform bacteria. But if you drink all 20 of the above described cups, you will definitely ingest exactly 1 coliform bacterium: never more, never less.

Comment

It is that precision that makes this a whole different situation. We can still ask statistical questions of the new situation, such as "How many cups can you drink and still have a good chance of not getting the bacterium?" The answer to such questions is not a number, it is a probability. But these are no longer rate problems, and they are dealt with in a different Lesson. The difference is worth a little reflection, and a little reflection is recommended herewith. The hard thing in statistics is not the formulas, but knowing which formula to use. Here before us is a typical decision point about which formula to use.