HOPES AND CHANCES. 729 



represents a fraction very near to that which has for a numerator the real 

 number of white balls existing in the urn, and for the denominator the total 

 number of balls. In other words, the ratios of the number either of extracted 

 white balls, or the whole of the white balls to the total number, tend to 

 become equal ; that is, the probability derived from this experiment 

 approaches indefinitely towards a certainty. The two fractions may differ 

 from each other as little as possible, if we increase the number of draws. 

 From this theorem we deduce several consequences. 



1. The relations of natural effects are nearly constant when these 

 effects are considered in a great number. 



2. In a series of events indefinitely prolonged, the action of regular and 

 constant causes affects that of irregular causes. 



Applications. The combinations presented by these games have been 

 the subject of former researches regarding probabilities. We will complete 

 our exposition with two more examples. 



Two persons, A and B, of equal skill, play together on the understand- 

 ing that whichever beats the other a certain number of times, shall be 

 considered to have won the game, and shall carry off the stakes. After 

 several throws the players agree to give up without finishing the game ; and 

 the point then to be settled, is in what manner the money is to be divided 

 between them. This was one of the problems laid before Paccal by the 

 Chevalier de Mere*. The shares of the two players should be proportional 

 to their respective probabilities of winning the game. These probabilities 

 depend on the number of points which each player requires to reach the given 

 number. A's probabilities are determined by starting with the smallest 

 numbers, and observing that the probability equals the unit, when player A 

 does not lose a point. Thus, supposing A loses but one point, his chance is 

 1*2, 3*4, 7*8, etc., according as B misses one, two, or three points. Supposing 

 A has missed two points, it will be found that his chance is as 1*4, i'2, 

 1 1*6, etc., according as B has missed one, two, or three points, etc. Or we 

 may suppose that A misses three points, and so on. 



We should note, en passant, that this solution has been modified by 

 Daniel Bernouilli, by the consideration of the respective fortune of the 

 players, from which he deduces the idea of moral hope. This solution, 

 famous in the history of science, bears the name of the Petersburgh problem, 

 because it was made known for the first time in the " Memoires de 1' Academic 

 de Russie." 



We will now describe the game of the needle. It is a genuine mathe- 

 matical amusement, and its results, indicated by theory, are certainly cal- 

 culated to excite astonishment. The game of the needle is an application 

 of the different principles we have laid down. 



If we trace on a sheet of paper a series of parallel and equi-distant 

 lines, AAI, BBI, cci, DDi, and throw down on the paper at hazard a per- 

 fectly cylindrical needle, a b, the length of which equals half the distance 



