ON DE MOIVRE's DOCTRINES OF CHANCE. 6/ 



worthy of a place in your valuable publication, their inser- chance ques- 

 tion will, I hope, elicit from yourself, or some one of your 

 mathematical readers, a few words in reply, to convince or 

 to confute a mind on all occasions suspicious of its own de- 

 ductions, whenever these deviate from the opinions of others, 

 old and learned in their walks of science. 



The celebrated de Moivre, in his woik. Case the first, 

 assumes the die as a familiar and favourable subject for de- 

 monstration : let us follow him in sense, though without 

 the advantage of his identical words, as I have not a copy 

 at hand. 



*' Any one undertaking, with a die of 6 sides, to cast an Chance of a 

 ^1 1 ,^1^ 111 • , ■ ^ throw wiih a 



ace in one throw, has A- of the 6 possible chances in his fa- sin<de die. 



vour, and the remaining f against him ; the whole 6 



chances being certainty, or at least such in the event of 



continued trials." Granted—' 



" Any one undertaking to cast an ace in two throws Chance of two 

 of one die, has for the first probability ^, as proved : ^'"'^*** 

 should the first fail, then the second remains, which is ^ 

 likewise; but the chance of the first failing is i, as that of 

 its succeeding is i; therefore the second throw is only ^ of 2d chance I 

 -^- for its chance of success, which added to the chance of ^^^ '^" ^* 

 casting an ace the firet throw, is ^ + i- of f — i- + ^^ r: -/^ 

 + ^-^yZZy^; the first throw being /-, the second only -g-*^." 



This doctrine I cannot grant. Nothing can prevent him But all the 



of the second throw, except his succeeding in the first ; P^'*^"^"^'J^® 

 .„. . equal to the 



therefore, either he has no occasion for it, or he has it in all whole. 



its full force and virtue of i chance, from which no circum- 

 stance can deduct. Otherwise it must be denied, that two 

 equal chances are twice as good as one; and by summing 

 up, according to de Moivre's rules, the probability of casting 

 an ace in six throws of one die, it will of course be found, 

 that they are below f, to which they should of course 

 amount, being the assumed sum of certainty on the event, 

 upon an average of trials. 



The chance of throwing a head with a halfpenny in two Instance inthe 

 throws, according to de Moivre, is, for the first, f; for the ^"g,^''^ * ^''^" 

 second, only | of ^ : so that the one is twice as good as the ' 

 other, and together they are ^ short in probability of what - 

 •was assumed certainty, or the amount of all chances, 



F 2 Any 



