OPINION AND PROBABILITY 281 



a priori probability. 1 According to his researches, if, for instance, 

 the antecedent probability of an event is f &quot; the odds are 1000 to 

 I that in 25,500 trials the event shall occur not more than 1 5,841 

 times, and not less than 14,819, that is, that the deviation from 

 15,300, the ideally probable number, shall not exceed -^ of the 

 number of trials&quot;. 2 Or, again, if from an urn containing white 

 arid black balls in the proportion of two white to one black, we 

 were to draw 90,000 times, returning each ball after drawing it, 

 the number of blacks drawn should be between 29,400 and 30,600, 

 showing a deviation of not more than 600, that is ?%%$?;&amp;gt; r TTTT&amp;gt; 

 from s&amp;gt; which is the antecedent probability. Were we, how 

 ever, to increase the number of draws 100 times (to 9,000,000), 

 the deviation should increase only ten times, i.e. to 6000, the 

 number of blacks drawn lying between 2,994,000 and 3,006,000 

 which would be a deviation of only J^TT frorn the antecedent 

 probability. 



We see, then, that as the number of experiences is multiplied, 

 the absolute magnitude of the fluctuations also increases, but 

 the amount of their difference from the antecedent probability 

 diminishes. 3 The first of those two facts enables us to foretell 

 the certain ruin, in the long run, of any one who continues to play 

 at a fair or equal game of chance. Suppose, for example, that 

 two persons, A and S, play at tossing pennies one at a time, each 

 player putting a penny stake on each toss, A betting for head, 

 and B for tail, throughout. While A is very wealthy, however, B 

 has only a shilling. The latter will be financially &quot; ruined &quot; when 

 the number of heads exceeds the number of tails by twelve. Now 

 the probability that this will take place within a certain number 

 of throws can be calculated, and will be found to increase steadily 

 and to tend ever towards certainty according as the total number 

 of throws increases. From the second part of the theorem 

 that the actual results tend progressively to approximate towards 

 the antecedent probability we infer that the greater the number 

 of experiences the greater will be the value of the results in point- 



1 Cf. CROFTON, art. on &quot; Probability &quot; in the Encyclopaedia Britannica ; 

 BAUDOT, art. on &quot; Probability &quot; in the Nouveau Dictionnnire des Sciences. 



JOYCE, ibid., p. 377. 



3 &quot; Le theorme de Bernoulli; revient a ceci : Si on fait un nombre inde&quot;finiment 

 croissant d 6preuves, 1 ^cart est infiniment petit par rapport au nombre des e&quot;preuves . 

 II faut bien remarquer quec est 1 e cart relatif qui tend vers ze&quot;ro, l e&quot;cart absolu devient 

 au contraire de plus en plus grand. &quot;BAUDOT, Probabilite ; apud JOYCE, ibid., p. 

 377. &quot; a. 



