KiDD. — On Prohahility. liii 



(1.) Tlie event A occurred in the first decade of June j or, 



(2.) It occurred in the second decade j or else, 



(3.) It occurred in the third decade. 

 That the event occurred on a day of June is assumed to be "known : the 

 numerical representative of this proposition is, therefore, the integer 1 3 and, 

 consequently, as each of the three alternatives constituting the case is equally 

 probable, the probability of each is represented by the fraction J. But the 

 proposition in question, viz., that the event occurred before the 21st day of 

 June, is true, if either of the first two alternatives be true ', while it is false, if 

 the third alternative be the true one. The probability of the proposition, 

 therefore, as furnished by those data, is §. 



The following familiar example is noticeable as presenting an easy problem, 

 in the solution of which a celebrated mathematician erred, and as illustrating, 

 accordingly, the fundamental distinction that there is between Probability and 

 Belief. Let us suppose a coin to be taken such that, when it is thrown, either 

 side of the coin is equally likely to fall uppermost. The probability of the 

 obverse side falling uppermost in any throw is J j for the case is comprised in 

 two equal alternatives, one of which affirms, and the other negatives, the 

 proposition in question. Let us now further ask. What is the probability of 

 the obverse being shown in one or other of two throws ? This case consists of 

 the following four alternatives : — 



(L) Both throws show reverse j or 



(2.) The first throw shows reverse, and the second obverse ; or 



(3.) The first throw shows obverse, and the second will show reverse ; or 

 else 



(4.) The first throw shows obverse, and the second also will show obverse. 

 These alternatives being all equally probable, and their number being 4, while 

 it is assumed that one or other of them is certain to occur, the probability of 

 each alternative is J. But any one of the last three alternatives affirms the 

 proposition in question ; so that the probability of the proposition is f . This 

 implies, that if the experiment were repeated a large number of times, the 

 obverse would be shown by one or other of two throws in three-fourths of the 

 instances or thereabouts ; and such experiments have actually been made with 

 results correspondent to the theory. But the eminent encyclopedist of the last 

 century, D'Alembert, inferred from the same data, that the resultant proba- 

 bility is f . He did not take into consideration that, in order to obtain the 

 accurate estimate of a probability, there must be equal quantities represented 

 by the units of the denominator. We may, of course, divide the case into only 

 the three most obvious alternatives, viz., those of reverse in each throw, 

 reverse in the first throw and obverse in the second, and obverse in the first 

 throw j but these alternatives do not present equal claims upon our acceptance. 



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