OPINION AND PROBABILITY 279 



its being drawn the first time would be , the second time | x |, 

 . . . the eighth time (f) 8 , or less than \\. If, therefore, it has not 

 appeared in eight draws, the probability that there is no such ball 

 in the urn is over 24 to I, which would generally be regarded 

 as practical certitude. 



Again, suppose it known that the three balls are either black or white, or 

 a combination of black and white in an unknown ratio. This gives four pos 

 sible alternative contents before the drawings commence, w w w, w w t&amp;gt;, w b 6, 

 b b b ; from which we gather that the probability in favour of a particular con 

 tent of the urn is \. Now if a white ball be drawn first, it excludes the possi 

 bility of b b b. As the remaining three alternatives show six possible ways of 

 drawing a white ball, the probability in favour of drawing any individual white 

 ball is \. Therefore the probability that the white actually drawn was from 

 ivivw, is or i, that it was from wwb, \ or ^, that it was from w b l&amp;gt;, I. That 

 is how the probabilities stand at the end of the first drawing. If, however, a 

 second drawing brings forth a black ball, w w u&amp;gt; is excluded, and the alterna 

 tives wwb and w b b are regarded as equally probable. But if a third drawing 

 gives a white, it makes the probability of wwb twice as great as that of tub b, 

 and its absolute probability therefore |. For, assuming wwbio be real, the 

 probability of securing the result actually attained, i.e. of drawing a white, a 

 black, and a white, is I x \ x | = 7 * T ; whereas the total probability of such a 

 combination from zy is J x -5 x J = ^y, or half as great as the former. 



The problem of estimating inverse probability of determin 

 ing to which of a number of possible alternative antecedents, or 

 groups of antecedents, a given event or series of events is due 

 is identical with the problem of determining the true magnitude 

 of a phenomenon by a series of approximate measurements (246, 

 268) ; for that problem might be stated thus : &quot; Given a series of 

 registered approximate measurements of a magnitude, what is 

 most probably the true magnitude that has yielded those measure 

 ments &quot;? 



When it is said that the probability of heading a coin is , or 

 of turning up a definite face in casting a die, , does this imply any 

 further certitude in regard to our data, beyond that already re 

 ferred to (265) the certitude that some face of the coin, or of the 

 die, will turn up at each throw ? It does not, about any individual 

 throw ; but it does imply further certitude about the nature of the 

 average of an indefinite number of throws : granted that all the 

 alternatives are and remain equally probable, we are certain that 

 in an indefinite number of throws the coin would be headed on an 

 average every second throw, or a definite face of the die turned 

 on an average every sixth throw. In other words, we are certain 

 that the a priori probability is in one case, and \ in the other. 



