Again: A group of integers is taken at random; show that the number thus taken is more likely to be odd than even. What we do in answering this is to start with any finite number n, and show that of all the possible combinations which can be made within this range there are more odd than even. Since this is true irrespective of the magnitude of n, we are apt to speak as if we could conceive the selection being made at random from the true infinity contemplated in numeration. 8. Where these conditions cannot be secured then it seems to me that the attempt to assign any finite value to the probability fails. For instance, in the following problem, proposed by Mr J. M. Wilson, "Three straight lines are drawn at random on an infinite plane, and a fourth line is drawn at random to intersect them: find the probability of its passing through the triangle formed by the other three" (_Ed. Times_, Reprint, Vol. V. p. 82), he offers the following solution: "Of the four lines, two must and two must not pass within the triangle formed by the remaining three. Since all are drawn at random, the chance that the last drawn should pass through the triangle formed by the other three is consequently 1/2." I quote this solution because it seems to me to illustrate the difficulty to which I want to call attention. As the problem is worded, a triangle is supposed to be assigned by three straight lines. However large it may be, its size bears no finite ratio whatever to the indefinitely larger area outside it; and, so far as I can put any intelligible construction on the supposition, the chance of drawing a fourth random line which should happen to intersect this finite area must be reckoned as zero. The problem Mr Wilson has solved seems to me to be a quite different one, viz. "Given four intersecting straight lines, find the chance that we should, at random, select one that passes through the triangle formed by the other three." The same difficulty seems to me to turn up in most other attempts to apply this conception of randomness to real infinity. The following seems an exact analogue of the above problem:--A number is selected at random, find the chance that another number selected at random shall be greater than the former;--the answer surely must be that the chance is unity, viz. certainty, because the range above any assigned number is infinitely greater than that below it. Or, expressed in the only language in which I can understand the term 'infinity', what I mean is this. If the first number be m and I am restricted to selecting up to n (n > m) then the chance of exceeding m is n - m : n; if I am restricted to 2n then it is 2n - m : 2n and so on. That is, however large n and m may be the expression is always intelligible; but, _m being chosen first_, n may be made as much larger than m as we please: i.e. the chance may be made to approach as near to unity as we please.