712 APPENDIX 



A common and important class of statistical deductions, which should 

 receive very critical examination, may be illustrated as follows : 



Suppose that, out of a total of ten years which have been observed, the 

 apple crop in this locality has been injured by frost four years, and has 

 been uninjured six years. If this data for ten years is all the evidence we 

 have bearing on the probability of an apple crop, the best estimate we can 

 give for the probability that the apple crop in this locality will not be 

 injured by frosts in a given year is T V If, however, our data extend over 

 twenty-five years, in fifteen of which the apple crop has been uninjured by 

 frost, we again give ^ (f = j%) as the best estimate for the probability 

 that the apple crop in this locality will not be injured by frosts in a given 

 year ; and certainly more confidence can be placed in the result than when 

 only ten years were taken. 



We might carry our illustration back a hundred years in sixty of which 

 the apple crop in this locality has been uninjured by frosts and we should 

 still give T % as the most probable value of the probability that an apple 

 crop in this locality will not be injured by frost in a given year. It should 

 be noted that we are here dealing with the probability of a probability, or 

 what De Morgan has called the " presumption of a probability." 



The critical examination of such probabilities as the above derived 

 from observation should include some criterion which will indicate the 

 accuracy of the approximation when only a limited number of cases can be 

 examined. Such a criterion may be found in the probable error of the 

 probability. 



The problem in hand may well be stated in the following general form : 



A bag contains an indefinitely large number of white and black balls in 



unknown ratio ; if m + n balls have been drawn as a random sample, and 



m are white and n are black, we give as the best value of the probability 



of drawing a white ball . What is the probable error in this result ? 



m + n 



Or, in other words, in m + n trials, an event has happened m times and 

 failed to happen n times ; if we deduce from this that is the probability 



that the event will happen on a given occasion, what is the probable error 

 in this result ? 



From the works of Laplace, Poisson, and De Morgan, it follows that 



m 0.6745 / mn 



the probable error in is given by the formula - \ 



m + n m + n \ m + n 



Applied to our illustration of the apple crop when the data covered only 

 ten years, this probable error formula gives yl = 0.104. 



From the magnitude of this probable error, it is at once seen that the 

 result T%- (derived from ten observations) for the probability that an apple 

 crop will be uninjured by frost can at most be said to be but a crude and 



