Karl Pearson Quotes

Monte Carlo roulette, if judged by returns which are published... is, if the laws of chance rule, from the standpoint of exact science the most prodigious miracle of the nineteenth century.

No statements of mere averages... are of the least avail against our statistics. Fluctuations from the averages are the sole reliable test...

The absence of the improbable, the redundancy of the probable, is just as much conclusive evidence against conformity with scientific law as the too frequent occurrence of the improbable itself.

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.

I felt somewhat taken aback. ...to have hit upon a month of roulette ...so improbable ...that it would only occur on the average once in 167,000 years of continuous roulette playing.

The next point to which I turned my attention was the frequency with which the several numbers themselves occurred. ...Each number might be expected to have occurred either 447 or 448 times. ... I found that they fitted to a standard deviation of 15.85 while the theoretical standard was 20.87 giving a difference of 5. ...What is a reasonable amount for the standard deviation of an experiment of this kind to differ from its theoretical value..? The mathematician answers... by finding the standard deviation of the standard deviation. It turned out... to be 2.43... the odds against a divergence as large or larger than 5...were ...21 to 1. In every two years I might expect such a deviation from the most probable results to occur once. ...I ...increased ...by counting the numbers for each week in the month instead of the total month. Here the experimental standard deviation [was] 7.2, the theoretical being 10.34, a difference of 3.14, while the standard deviation between experiment and theory was only 0.60. The odds against a divergence so great as this are... about 2,000,000 to 1.

[W]e find out of 16,019 trials, 8053 red numbers instead of 8009 or 8010. We have a deviation of 43 to 44. The standard deviation is about 63; a deviation as great as or greater than 44 would occur in about half the number of times in which 16,019 returns were examined. It presents therefore nothing of the remarkable or improbable.

[F]or every type of experiment there is a numerical quantity, depending partly on the chance of the single event succeeding, and partly on the total number of the trials of it, which we may term the . ... it gives us a measure of the frequency with which deviations of various sizes will occur.

[W]e have an equal number of black and red possibilities... Thus in a very great number of throws there ought to be 50 per cent of both. ...In no case... are the results exactly reached, but in all the cases of large numbers we have but small deviations from 50 per cent. Thus 16,141 roulette throws give slightly better results than 12,000 and slightly worse results than 24,000 tosses. We notice that... 16,019 roulette throws give nearly the worst percentage 50.27 instead of 50.

[R]andom spinning being assumed, the distribution of chance in the game depends upon the mechanical perfection of the teetotum; it must be equally likely to fall on all its thirty-seven sides, i.e. the frequency of all the numbers must in the long run be very nearly the same.

[T]he records of the tables are published in... Le Monaco and issued weekly in Paris... eight weeks roulette gave a grand total of 33,000 events. ...M. Blanc deserved a niche in the temple of science and Le Monaco a shelf in every mathematician's library.

In my enthusiasm Monte Carlo appeared to me in a new light; it was clearly a scientific laboratory preparing material for the natural philosopher.

25,000 tosses of a shilling occupied a good portion of my vacation and... gave me... a bad reputation in the neighbourhood... A friend and former pupil supplemented... 8200 penny trials and the drawing of 9000 tickets from a bag while another kindly provided... nearly 23,000 drawings of coloured and numbered counters. In all these cases the results were in... strikingly close agreement with theory.

The scientific conception of chance is that of a measure based on experience; a knowledge of the average results of many events is used to replace ignorance of the result of any individual event. ...The judgment which Science gives in this case is decisive; judged by the so called "permanences," or runs of colour, Monte Carlo roulette is no true worship of the goddess at all.

While the moralists have boldly asserted that the service of the goddess Chance leads to a complete demoralisation of her worshippers... the mathematicians... start from the hypothesis that gambling is a game of chance—chance being defined in their own perfectly clear and definite sense. My object... is to show that chance in this sense... as it applies to the tossing of an unloaded coin, has no application to Monte Carlo .