ON PROBABILITY. ; 
Sn the 9 5 
Rass Jas = 31. In the same manner, if he throw a deuce 
vork on {F 
Wasli the first time, he cannot, in the two throws which this secures to him, score 
less than two or more than 12, and it is easy to see that the chances of each 
0 an number equidistant from the mean throw, 7, are equal; 7 is, therefore, his 
es Je expectation on this supposition. If 3,4, 5, or 6 be thrown the first time, 
pag, his expectation on each will similarly be found to be the means between 3 
Riche and 18, 4 and 24, 5 and 30, 6 and 36, or 10%, 14, 17%, and 21 respectively, 
deny) so that the expectation of his throws will be 1 {31 7, + 10%, + 14 
lontmo + 17% + 21 } = 12% points, from which it would appear that the odds dre 
ocation in his favour. From the manner in which Bernoulli dwells upon the plau- 
Ty sibility of this solution, it seems not improbable that he had been himself 
ied on deceived in the first instance, by the erroneous view which he here exposes 
i” Th to his readers. The error consists in not multiplying each chance separately 
id Dts by the gain or loss it could occasion to the player. The fourth part of Ber- 
ve fo an noulli’s book, which had been expected with the greatest impatience, but 
sient top. which unfortunately was left incomplete at the time of the author's death, 
hstion was intended to contain an application of the theory of probabilities to the 
_ examination of questions connected with civil and domestic life, Imperfect 
bi the sgh. as this has been left, it must, undoubtedly, be considered as the foundation 
Detioive of whatever has been since done in this branch of the science. Bernoulli 
ny troduce the term © 1 tainty,” 
Deion seems to have been the first to intro e term “moral certainty, on 
biz which we have already remarked in the body of the treatise. He also, in 
imitation of Aristotle,* distingnishes between what he calls free and casual 
Ne contingencies, classing under the former all those contingent events which 
oo] depend on the will of a rational being. He also inculcates strongly the 
”. fundamental principle, from the neglect of which so much error and confu- 
3 sion have arisen, that “contingency or chance has reference merely to the 
fnyghe state of our knowledge.” After explaining the principal rules by which we 
eond op should be guided in our investigations, he proceeds, among other things, to 
nd elegy examine the method of determining probabilities a posteriors, that is to say, 
vers of | by often-repeated experiment: and shows, in the noted theorem which still 
enarabl bears his name, that the probability of attaining to the knowledge of the 
its Nom probability of an unknown .event constanfly increases with the number of 
li. part of experiments made upon it, so that we can always, by multiplying our expe- 
ated Latin riments, reach a degree of probability as near certainty as we choose to fix 
mited to upon, that the error of our estimation lies within given limits. With this 
It does proposition the work abruptly terminates. At the end of Bernoulli’s book is 
sis, Which an anonymous letter on the game of tennis, the author of which is not cer- 
t gives the tainly known, but the theory, language, and notation, strongly mark it to 
he follow- belong, if not to James Bernoulli himself, at any rate to some one trained 
e, though in the same school, and fully imbued with his ideas and opinions. 
e says, 10 80. The first great step beyond what Bernoulli had suggested, showing the 
hese ues use to be made of experiments in estimating unknown probabilities, is con- 
tes 47 tained in a posthumous paper by Bayes, inserted in the Philosophical Trans- 
bo actions for 1763, through the means of Dr. Price, so well known by his own 
publication on the subject of annuities. The problem .which Bayes pro- 
I poses to solve in this paper is the following :—Given the number of times in 
which an unknown event has happened and failed. Required the chance 
Jf that the probability of its happening in a single trial lies somewhere between 
any two degrees of probability that can be named. When disencumbered of 
: the geometrical form under which it was then the fashion to represent inte= 
nally 8 # Physica, lib. ii, cap, 6, 
J 
i