The next point to which I turned my attention was the frequency with which the several numbers themselves occurred. ...Each number might be expected … - Karl Pearson

I determined to... investigate how closely the runs, that is, successions of numbers of the same colour were in accord with theory. ...The chance of a head<math>=\frac{1}{2}</math>, of two heads succeeding each other <math>\frac{1}{2}\times\frac{1}{2} = \frac{1}{4}</math>, of three heads <math>\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2} = \frac{1}{8}</math>, and so on. Calling a "set" the run of tosses or the throws of the roulette ball till a change of face or of colour comes, the chance of a change<math>=\frac{1}{2}</math>, of a persistence followed by a change <math>\frac{1}{2}\times\frac{1}{2} = \frac{1}{4}</math>, and so on. ...[I]n the case of the roulette on one occasion the actual deviation is nearly ten times the standard.... The odds are thousand millions to one against such a deviation as nine or ten times the standard. ..My pupil... tabulated... the runs in a second fortnight's play with the result... so improbable that it was only to be expected once in 5000 years of continuous roulette. ...Finally, Mr. de Whalley investigated 7976 throws of the ball, forming a fortnight's play, at a slightly later date... There resulted deviations 4.63, 4.62, and 4.44 times the standard deviation, or odds of upwards of 263,000 to 1... That one such fortnight of runs should have occurred in the year 1892 might be looked upon as a veritable miracle, that three should have occurred is absolutely conclusive. Roulette as played at Monte Carlo is not a scientific game of chance.

The purpose of the mathematical theory of statistics is to deal with the relationship between 2 or more variable quantities without assuming that one is a single-valued mathematical function of the rest. The statistician does not think a certain x will produce a single-valued y; not a causative relation but a correlation. The relationship between x and y will be somewhere within a zone and we have to work out the probability that the point (x,y) will lie in different parts of that zone. The physicist is limited and shrinks the zone into a line. Our treatment will fit all the vagueness of biology, sociology, etc. A very wide science.

Wherever there is the slightest possibility for the human mind to know, there is a legitimate problem of science. Outside the field of actual knowledge can only lie a region of the vaguest opinion and imagination, to which... men too often... pay higher respect than to knowledge.