5 As by Prévost in the Bibliothèque Universelle de Genève, Oct.[ TN: space] 1829. The explanation is noted, and apparently accepted, by Quetelet (Physique Sociale, I. 171).
6 Essay on Probabilities, p. 126.
7 This theoretical or absolute neglect of what is very rare must not be confused with the practical neglect sometimes recommended by astronomical and other observers. A criterion, known as Chauvenet's, for indicating the limits of such rejection will be found described in Mr Merriman's Least Squares (p. 166). But this rests on the understanding that a smaller balance of error would thus result in the long run. The very rare event is deliberately rejected, not overlooked.
8 The process of calculation may be readily indicated. There are, say, about 350,000 letters in the work in question. Since any of the 26 letters of the alphabet may be drawn each time, the possible number of combinations would be 26350,000; a number which, as may easily be inferred from a table of logarithms, would demand for its expression nearly 500,000 figures. Only one of these combinations is favourable, if we reject variations of spelling. Hence unity divided by this number would represent the chance of getting the desired result by successive random selection of the required number of 350,000 letters.
If this chance is thought too small, and any one asks how often the above random selection must be repeated in order to give him odds of 2 to 1 in favour of success, this also can be easily shown. If the chance of an event on each occasion is 1/n, the chance of getting it once at least in n trials is 1 − (n − 1/n)n; for we shall do this unless we fail n times running. When (as in the case in question) n is very large, this may be shown algebraically to be equivalent to odds of about 2 to 1. That is, when we have drawn the requisite quantity of letters a number of times equal to the inconceivably great number above represented, it is still only 2 to 1 that we shall have secured what we want:—and then we have to recognize it.
9 The longest life which could reasonably be attributed to any language would of course dwindle into utter insignificance in the face of such periods of time as are being here arithmetically contemplated.
10 We are here assuming of course that the ultimate limit to which our average tends is known, either from knowledge of the causes or from previous extensive experience. We are assuming that e.g.[** TN: space] the die is known to be a fair one; if this is not known but a possible bias has to be inferred from its observed performances, the case falls under the former head.
11 Except indeed the gamblers. According to a gambling acquaintance whom Houdin, the conjurer, describes himself as having met at Spa, “the oftener a particular combination has occurred the more certain it is that it will not be repeated at the next coup: this is the groundwork of all theories of probabilities and is termed the maturity of chances” (Card-sharping exposed, p. 85).