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A Treatise on Probability. By J. M. Keynes. Fellow of King's College, 

 Cambridge. [Pp. xi + 466.] (London : Macmillan & Co., 192 1. 

 Price i8s. net.) 



The subject of probability is one which has attracted the attention of many 

 famous mathematicians. Some have given it only casual notice, whilst others 

 have devoted much time to it. The connection with games of chance is 

 doubtless responsible to some extent for the wideness of its appeal. The 

 mathematician has been frequently tempted, however, to regard probability 

 as a purely mathematical conception and to neglect the logical basis upon which 

 it is fundamentally based. The neglect of this aspect has been responsible 

 for some of the confusion and errors into which mathematicians have at times 

 been led. Probability must in any case be relative to certain premisses, and a 

 modification of these will modify the value of the probability. Thus to take a 

 simple example quoted by Mr. Keynes : " If a chord in a circle be drawn at 

 random, what is the probability that it will be less than the side of the in- 

 scribed equilateral triangle ? " On the assumption that it is indifferent at what 

 point one end of the chord lies, the probability is | : if it is indifferent in what 

 direction the chord lies the probability is ^, whilst if it be supposed that the 

 middle point of the chord is chosen at random, the probability is J. The prob- 

 ability is therefore indeterminate without a more precise definition of the 

 initial premisses. 



Another example in which it is not at first sight quite so apparent that the 

 initial premisses are not complete is the following. Mathematicians have 

 generally assumed that the probability of two witnesses speaking the truth, 

 who are independent in the sense that there is no collusion between them, is 

 the product of the probabilities that each of them separately will speak the 

 truth. If, therefore, x and y are the probabilities that two independent wit- 

 nesses X and y speak the truth, and if they both agree in a particular statement, 

 the probability that the statement is true is generally taken to be xyjxy + 

 (i — x) (i — y). For xy is the chance that they both speak the truth and 

 (i — x) (i — y) the chance that they both speak falsely. A critical examina- 

 tion of this problem by Mr. Keynes shows that herein it is tacitly assumed that 

 any answer to a given question is, a priori, as likely as not to be correct, which 

 cannot in general be the case. 



As a further example, of a somewhat different nature, of the errors into 

 which it is so easy to be led, we may instance Bernouilli's Theorem, frequently 

 used in problems of statistical inference. This theorem asserts that, if the 

 probability of an event's occurrence under certain conditions is p, then if 

 these conditions are present on m occasions the most probable number of the 

 event's occurrence is mp (or the nearest integer to this), i.e. the most probable 

 proportion of its occurrence to the total number of occasions is p. This 

 theorem has been regarded by some as having universal validity. Mr. Keynes 

 shows, on the other hand, that it holds only under certain conditions, which 

 are generally not fulfilled. For the theorem to be valid, the initial data must 

 be of such a character that additional knowledge as to the proportion of 

 failures and successes in one part of a series of cases is altogether irrelevant 

 to our expectation as to the proportion in another part. This is rarely the 

 case, for the initial probability is generally founded upon experience, and it is 

 liable to modification in the light of further experience. 



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