By Charles L. Dodgson, M.A.
In the years 1885, 1886, there appeared in regard to a Solution of Quest. 7695 (see Vol. xliii., p. 86, and xliv., p. 24) a discussion about a difficulty in the Theory of Chances, of which the following question was treated as a typical example:—“A random point being taken on a given line, what is the chance of its coinciding with a previously assigned point?” On one side it was maintained that the chance is absolute zero: on the other side it was maintained, by myself and others, that it is some sort of infinitesimal, and not absolute zero. The arguments on both sides were fully stated, and my only excuse, for re-opening the discussion, is that I have a new view of the difficulty to offer to the supporters of the “absolute zero” theory.
I assume that both sides accept the following axioms:—(1) that no aggregate, however infinitely numerous, of absolute zeroes can constitute a magnitude, however infinitely small; (2) (an example of the preceding) that no aggregate, however infinitely numerous, of points can constitute any portion, however infinitely short, of a line; and hence (3) that, if the chance of a random point on a line coinciding with a single selected point be absolute zero, so also is its chance of coinciding with one or other of a selected aggregate of points, however infinitely numerous.
I now propose two questions:—
I. “A random point being taken on a given line, what is the chance of its dividing the line into two commensurable parts?” It seems clear that we are here dealing with a selected aggregate of points, since it is impossible to mark off any portion of the line, and to say “Wherever, in this portion, the random point shall fall, it will divide the whole line into two commensurable parts.” I assume, then, that my opponents would answer “It is absolute zero.”
II. “And what is its chance of dividing the line into two incommensurable parts?” Here again they must answer “It is absolute zero.”
And yet one or other of these two events must happen! Hence, the sum of the two chances must be mathematically represented by unity; that is, one or other (though we cannot say which) must be—not only “something,” not only a certain infinitesimal, of some inconceivably high order—but must actually reach, if not exceed, the finite value of one-half!