Mr. F. Y. Edge worth's Problems in Probabilities. 381 



legislators in 1844, when they fixed the limit of uncovered 

 Bank of England notes at £14,000,000; upon the ground 

 that the Notes in the hands of the public (less* by the Bank 

 Post Bills together with the lost notes) had never fallen below 

 that figure. What is the probability that if M is the maxi- 



mum or minimum (measured from the Mean) of -x observa- 

 tions, a subsequent observation, the (^ + l)thin excess (or 



defect), will not exceed M, or more generally ^M. If, like 

 the legislators, we are to ignore the grouping of the n obser- 

 vations, and to utilize only the datum that M is the maximum 

 of n observations on one side of the Mean ; then a priori one 

 ratio of the given maximum to the unknown modulus may be 

 regarded as about as likely as another. Consider the parti- 

 cular ratio r. The probability that a single observation in 

 excess of the Mean will fall within M is 6(r); where, as before, 



2 C x 

 6(oc) is identical with -y= I e~ t2 dt. The probability that n 



V v Jo 

 observations being in excess of the Mean should not exceed 

 M is [0(r)~\ n . Hence the a posteriori probability that the par- 

 ticular hypothesis considered is the true one is 



[0(r)] +£' 



'0 



The probability that if the particular hypothesis is true the 

 next observation above the mean will not exceed ^M is 6(qM.) . 

 Hence the a posteriori probability that the next observation 

 will not exceed ^M is 



V [#M] n 0($r) dr -f- f °° [6(r)]\ 



J o Jo 



I have attempted roughly to evaluate f this expression for cer- 



n 

 tain interesting values of q and n. First, let ^ = ^0 and q — 1. 



We have then a problem much the same as that which the 



legislators in 1844 set to themselves. I find that the odds 



against the regulation limit £14,000,000^ being passed in the 



proximate future were below a hundred to one. A safer 



limit would have been constructed by putting q=-\\. The 



odds against this limit (£12,500,000§) being passed are some 



n 

 hundreds to one. Again, suppose ~ = 20. To obtain the 



* Above, p. 373. 



t By Tables analogous to that given above. 



X £15,000,000 with Bills and lost Notes ; see above. 



§ £14,000,000 with Bills and lost Notes. 



