TRANSACTIONS OF SECTION D. 



889 



dice. My wife has spent some time during the last two months in tossing dice for 

 you, and I will ask you to look at the results. 



Her first record gives the number of dice showing more than three points in 

 each of 4,096 throws of twelve dice. There are, of course, six numbers on each of 

 the dice ; so that if all the dice were perfectly symmetrical and similar, the average 

 number of dice with more than three points should be six in each throw of twelve. 

 But dice are not symmetrical and similar. The points on the dice used were 

 marked by little holes, scooped out of their faces ; and the face with six such holes 

 scooped out of it was opposite to the face with only one such hole : so that the 

 face with one point was heavier than the face with six points ; and therefore six 

 was rather more likely to be uppermost than one. In the same way, two was 

 opposite five ; so that the five face was a little more likely to fall uppermost than 

 the face with two points. Therefore, it is a little more likely that you will throw 

 four, five, or six, in throwing dice, than it is that you wUl throw one, two, or 

 three. 



Accordingly, the average number of dice, in these 4,096 throws, which had 

 more than three points, was not six, but 6'1.35. 



To show you that this excess of high points was due to some permanent pro- 

 perty of the dice, she threw these twelve dice another 4,096 times ; and the average 

 number of dice with more than three points was 6'139. A third series of trials 

 gave an average of 6- 104, and a fourth gave an average of 6-116. 



You see that the difference between the highest and the lowest of these deter- 

 minations is only about one-half per cent., so that the mean result of such a series 

 of fortuitous events can be determined with great accuracy. 



And just as the mean of the whole series can be determined, so we can know 

 with considerable accuracy how often any possible deviation from the average 

 result will occur. The degree of accuracy with which we can know this may be 

 judged from Table I. 



Table I. — Frequency with which Dice showing more than three points ivere thrown 

 each of Four Series of Trials, the number of throics in each Series being 

 = 4,096. 



212. 



You see that the results of the experiments agree fairly well with one another, 

 and differ from the results moat probable with symmetrical dice, in the way which 

 the structui-e of the actual dice would lead one to expect. Throws which give 

 seven, eight, or nine dice with more than three points occur too often, throws in 

 which only two, three, or four dice have more than three points do not occur often 

 enough. You see then that each of these results is orderly and regular, and that 

 the four results agree very fairly among themselves, not only in the mean value of 

 each of them, but in the magnitude and frequency of departures from the mean. 



