Poisson Distribution

p(1 or more) = 0.3935, or about 40%.

Explanation

"At least 1 bacterium" means 1 or more bacteria, and for each of those options there will be a separate probability; we can get the answer by adding up all those probabilities, out to infinity. But there is a shortcut. We can look up the probability of zero bacteria in the same r = 0.5 column of the Poisson Table, which is p(0) = 0.6065, and subtract that result from 1 to get the answer given above.

Comment

If we take 1 sample with our 10cc test tube, the likeliest result (at 60%) is 0 bacteria. This is not a good representation of the true value. But if we take 10 samples of this size, we may see from the table that, in the most likely case, 6 of the samples will have 0 bacteria, 3 samples will have 1 bacterium, and probably 1 will have 2 bacteria. The total bacteria captured amount to 5, and the total volume of those samples is 100cc. The implied concentration in that case will be, not 0, but 5 per 100cc, or 1 per 20cc, which is the true answer. This is a demonstration in miniature that more samples (especially if the samples themselves are small) give a more accurate answer than 1 sample. The question of how big the sample should be, and how many of these samples should be taken, to give a reasonable account of the thing being sampled, is one on which a great deal of time has been spent. We will mention the results in another Lesson.

Meanwhile, it will be intuitively obvious that if only one 10cc sample is taken, the results are likely to be rather variable; this shows that the sample is not adequate to reflect the reality. If ten 10cc samples are collected as a group and the results noted, and if that experiment is repeated, and again repeated, and if the results of those larger samples are relatively stable, then we have reached a sample size which, whatever theory may suggest, is empirically more adequate to the nature of the thing being sampled, and we can be relatively content, as a practical matter, to stop at that level. The additional refinement of still larger sampling would not be worth the time and money it would cost. The thing in statistics is not to reach a perfect answer, which due to random variation at the heart of the universe we cannot do anyway, but to reach a good enough answer, and then quit.

It is no doubt vulgar and unphilosophical to mention these mundane considerations, but statistics without mundane considerations is merely another form of hot air.