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LXIII. On certain Questions relating to the Theory of Probabili- 

 ties.— ?2iXi II. By W. F. DoNKiN, M.A., F.R.S., F.R.A.S., 



Savilian Professor of Astronomy in the University of Oxford. 



1. TN a former paper I gave a brief sketcli of the general 

 A principles of the theory of probabilities considered under 

 the aspect which seemed most advantageous with reference to 

 its practical applications. I also discussed some problems with a 

 ^iew to exhibit the accordance between the results of the theory 

 and the conclusions of common sense. In continuing the sub- 

 ject, I mil begin by noticing the important distinction, to which 

 Professor Forbes has drawn attention, between what may be 

 called imjirobability and incredibility ; that is, between the diffi- 

 culty of believing beforehand that an assigned event will happen, 

 and the difficulty of believing afterwards that a recorded event 

 has happened. The distinction is perfectly well known, and re- 

 cognized in the mathematical theory ; but there are some points 

 connected with it which it is worth while to examine a little in 

 detail. Professor Forbes's experiment with the grains of rice 

 will fm-nish an illustration*. 



2. Let us suppose that n grains are thrown into a box divided 

 into m equal compartments. Also let n be a multiple of m (sup- 

 pose n=rm), so that an equal distribution amongst the compart- 

 ments is possible. Let a,b, c, . . . a, /3, y, . . . be whole numbers 

 such that a + b + c+ ... ^?h, and 



aoc + b^ + cy+ . . . =n. 



What is the antecedent probability that there will be a compart- 

 ments containing a. grains each, b compartments containing yS 

 grains each, &c. ? The antecedent probability of any given dis- 

 tribution in which each individual grain should be assigned to 

 a determinate compartment, is m~'\ The number of M'ays in 

 which the n grains can be divided into a + b + c+ . . . parcels, 

 whereof a parcels shall contain a grains each, b parcels shall 

 contain /3 grains each, &c. is 



1 ■ 2 . 3 . ■ . w 



(1.2.3...a)«.(1.3.3.../3)*....1.2.3...a.l.2.3...*...' ^ ^^ 



and the number of ways in which the parcels can be arranged in 



the m compartments is ?h(»z — l)(m — 2) . . . (??^ — s + 1), where 



s=a + b + c+ .. . Hence the probability required is found by 



,^. , . m{m—1)...{m—s + l), ,, • , i v -r 



multiplymg — ^ — ^~- — ^ by the expression (A.). It 



* I write wthout the opportunity of consulting Professor Forbes's paper 

 in the Philosophical Magazine ; my references to it will therefore be from 

 memory. 



