890 



REPORT — 1898. 



That they differ from the results which -would probably be obtained with symme- 

 trical and similar dice is only to be expected, because the dice used are neither 

 symmetrical nor similar. 



You notice that this table is very nearly symmetrical ; the most frequent 

 result is that which lies in the middle of the series of possible results : and the 

 other frequencies would, with perfect dice, be distributed symmetrically on each 

 side of it ; so that with perfect dice one would be as likely to throw five dice out 

 of twelve with more than three points as one would be to throw seven, and so on. 



This symmetry in the distribution of the results is only found when the chance 

 of the event occurring in one trial is even. The next table shows the result of 

 4,096 throws of twelve dice, in which sixes only were counted. The chance against 

 throwing six with any one of the dice is of course five to one ; so that in throwing 

 twelve dice you are more likely to throw two sixes than to throw any other 

 number. But you see that the chance of throwing only one six is very much 

 greater than the chance of throwing three ; the chance of throwing none is greater 

 than the chance of throwing four, and while there is a chance of throwing five, six, 

 or more, of course it is impossible to throw less than none at all ; so that the dia- 

 gram is all askew. You see that this time, as before, the frequency with which 

 any number of sixes did actually occur was as near to the result most probably with 

 perfect dice as the asymmetry of the actual dice allows one to expect.* 



Table II. — Frequency of Sixes in 4,096 tJiroios of Twelve Dice. 



These results will be enough to show you how absurd is the attitude which so 

 many writers have taken up towards chance when discussing animal variation. 

 The assertion that organic variation occurs by chance ia simply the assertion that 

 it obeys a law of the same kind as that which expresses the orderly series of re- 

 sults we have just looked at.'- 



That is a matter which can be settled by direct observation. But in order to 

 express the law of chance in such a way that we can apply it to animal variation, 

 we must make use of a trick which mathematicians have invented for that purpose. 



It is a well-known proposition in probability that the frequency with which one 

 throws a given number of sixes in a series of trials with twelve dice is proportional 

 to the proper term in the expansion of (^ + f)'"- The most probable numbers 

 in this table were calculated by expanding this expression. But if I had 

 wanted to show you the most probable result of experiments with 100 dice, I 

 should not -willingly have expanded (^ + f )^°°. The labour would be too enormous. 



' It is unfortunate that I chose dice as instruments in these experiments. Dice 

 are not only sensibly asymmetrical, but any ordinary dice are sensibly digsimilar ; so 

 that the result most probable with any actual dice is not given by a simple binomial 

 expansion. The result theoretically most probable for the actual dice used could 

 not be determined -without very careful measurement of the dice themselves ; and I 

 was unable to attempt measures of the requisite accuracy. All that the records 

 show, as they stand, is the amount of agreement between four successive observa- 

 tions of a fortuitous event. 



- The law is not, however, identical in the two cases ; see infra. 



