CHAPTER LVI. 



THE ANALYSIS OF CHANCE AND MATHEMATICAL GAMES. 

 MAGIC SQUARES THE SIXTEEN PUZZLE SOLITAIRE EQUIVALENTS. 



( WE will now proceed to draw our readers' attention to several experiments 

 very famous at a former period, but which our own generation has completely 

 overlooked. We refer to the Analysis of Chance, a science still known 

 under the title of Calculation of Probabilities, formerly cultivated with so 

 much ardour, but to-day almost fallen into oblivion. 



Originating in the caprice of the clever Chevalier de Mere, who in 

 1654 suggested the game to Pascal, the analysis of chance has given rise to 

 investigations of an entirely novel kind, and attempts have been made to 

 measure the mathematical degree of credence to be given to simple conjec- 

 tures. We will first recapitulate the principles laid down by Laplace on 

 this subject. We know that of a certain number of events, one only can 

 happen, but nothing leads us to the belief that one will happen more than 

 the other. The theory of chance consists in reducing all the events of the 

 same kind to a certain number of equally possible cases, such, that is to 

 say, that we are equally undecided about, and to determine the number of 

 cases favourable to the event, whose probability we are seeking. The ratio 

 of this number to that of all possible cases is the measure of this probability, 

 which is thus a fraction, the numerator of which is the number of favourable 

 cases, and the denominator the number of all possible cases. When all the 

 cases are favourable to an event, its probability changes to cer- 

 tainty, and it is then expressed by the unit. Probabilities increase or 

 dimmish by their mutual combination ; if the events are independent of each 

 other, the probability of the existence of their whole is the product of their 

 particular probabilities. Thus the probability of throwing an ace with one 

 dice being , that of throwing two aces with two dice is -f$. Each of the sides 

 of one dice combining with the six sides of the other, there are thirty-six 

 possible cases, among which one only gives the two aces. When two events 

 depend on each other, the probability of the double event is the product of 

 the probability of the first event by the probability that, that event having 

 occurred, the other will occur. This rule helps us to study the influence of 

 past events on the probability of future events. If we calculate d priori the 

 probability of the event that has occurred and an event composed of this 



