""Likely" refers to probability, and yet the poi..."


by Robert Eidschun Nov 14 2017 updated Nov 14 2017

Then each molecule of red water is 90% likely to make it to the shared pool, and each molecule of blue water is 30% likely to make it to the pool\. \(90% of red water and 30% of blue water make it to the bottom\.\) So each molecule of red water is 3 times as likely \(0\.90 / 0\.30 = 3\) as a molecule of blue water to make it to the bottom\.

"Likely" refers to probability, and yet the point of this essay is to explain probability. Therefore, the use of "likely" is, in a sense, circular reasoning. After all, what does "likely" mean? It's not explained here. It suggests an outcome frequency of sorts and so this statement and others like it is an attempt to arrive at an outcome frequency (equivalent to the proportions of red and blue water that make it down through) by referring to another outcome frequency; thus the circularity.

Better to stick with the proportions themselves by explaining that, however much red water makes it down through, there will be three times as much of it as there is blue water that makes it down through. Say that some fraction, f, of the blue water molecules makes it down through; then for every 100 molecules of water, f x 80 blue molecules make it down through and 3f x 20 red molecules make it down through, making for proportions of 60f red to 80f blue. Scaling down those proportions by dividing both by f, we get 60:80, which can be further scaled down to 3:4.

Note that the factor of 3, i.e. the "likelihood ratio" (by which the initial proportions of 20:80 are multiplied) is explicit in the previous paragraph. (It's in the statement, "3f x 20 red molecules make it down through".) Putting it another way, the previous paragraph makes it clear that multiplying by 3 will give the same final proportions ("posterior odds") as will, in taking a frequency approach, multiplying 20 by 0.9 and 80 by 0.3, since the latter proportions can be scaled by dividing each by 0.3: (0.9/0.3 x 20):(0.3/0.3 x 80) = (3 x 20):1 x 80 = 3:4.