Poisson Distribution
Answer 5

p(2) = 0.0758, or about 1 chance in 13.

Explanation

If the concentration (r) is 1 per 20cc, it is also 0.5 per 10cc, which is here the unit of observation. We can then immediately read off all probabilities from the 0.5 column of the Poisson Table. The table directly supplies the answer given above.

Comment

Study of the Poisson Table will reveal something important about sampling. For instance, we might expect that the chance of getting a reading twice as high as the expected value would be the same for the 20cc test unit (rate = 1, twice the rate = 2) as for the 10cc test unit (rate = 0.5, twice the rate = 1). The table shows that this is not true. The chance of an aberrantly high reading is less with the larger volume (0.1839) than with the smaller volume (0.3033). On reflection, this will seem natural. The rate, or concentration, in this problem is after all very small. It stands to reason that a small test volume will often not detect it, or may misrepresent it. A larger volume is more likely to contain a representative sample of material.

One of the great issues in statistics is how large the test unit has to be, or how many small-unit tests have to be made, to give a reasonably sure answer. The Poisson Table, reflectively studied, will suggest the texture of this problem. Answers to the problem are given elsewhere. The important thing is not to memorize formulas, but to develop an intuitive feeling for the nature of the problem to which they are the answer.

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