462 Prof. Donkin on certain Questions relatinc/ to 



in which they enter into the solution of a posteriori problems ; 

 an illustration which I should have thought unnecessary, if it 

 had not appeared that misconception on the subject is probably 

 not uncommon. 



9. It was observed above (art. 3), that the fallacy of confound- 

 ing a priori with a posteriori probabilities may happen in par- 

 ticular cases to produce very trifling numerical errors. For 

 example, let p be the a priori probability of an observed phseno- 

 menon, on the supposition of the existence of a cause A ; and 

 let it be supposed that if another cause B existed, which could 

 not coexist with A, the phfenomenon would certainly result. 

 Also suppose the a priori probabilities of the existence of these 



two causes to be each = ■^. Then the a posteriori probability of 



the existence of A is Y^^—, which is sensibly equal to ^, if jo be 



very small. Possibly some consideration of this sort, together 

 with the great complexity of the rigorous application of the 

 theory to the question of the grouping of stars, may have led 

 Sir J. Herschel to mention the subject in a way which seems to 

 coimtenance the fallacy alluded to. (Outlines of Astronomy, 

 p. 565.) 



10. I shall now proceed briefly to consider this problem, and 

 shall endeavour to show how a numerical estimate of the true 

 a postej-iori probabilities, which ought to be substituted for Mit- 

 chell's results, could be obtained supposing all mere difliculties 

 of calculation overcome. In order to make the problem at all 

 practicable, it is necessary to simplify it as much as is consistent 

 fldth retaining a tolerable similarity to the facts of nature. 



Let us suppose that there are n visible stars of a certain class, 

 such that if any two of them were within certain limits of an- 

 gular distance, no conclusion could be di-a\\ai from their apparent 

 brightness as to whether they fomied a true binary system, or 

 were merely optically double. And let us fui'ther suppose that 

 there are actually m pairs of stars within these limits of angular 

 distance, the remaining n—2m stars being all distinctly single. 

 The principal question will be, what is the probability, arising 

 from these phsenomena, that fx, out of the m pairs of stars are 

 real binaiy systems ? fi being any assigned number. 



11. Let the word system stand indifferently for single star or 

 binary system, and let p be the a priori probability that a pro- 

 posed system would be binary. We will suppose all systems but 

 single and binary excluded, so that 1 —p will then be the ujiriori 

 probabUity that a proposed system would be single. The mean- 

 ing of y; may be explained as follows. Svippose a person to be 

 perfectly acquainted with the mode in which stars are produced ; 

 he would be able, setting aside difficulties of calculation, to assign 



