MEMORIAL TO CLASSICAL STATISTICS 121 



The most probable distribution is the one evoked by the caller, when he 

 calls for an equal number of balls in every basket. If there are N balls and 

 M baskets, this means N/M balls to each basket, and a maximum number of 

 inventories which I will call irmax- Looking back to equation (1), we see 

 that irnuix is a fraction the numerator of which is A—factorial, while the de- 

 nominator is (iV/M)-factorial raised to the power M. Taking logarithms 

 and using the super-Stirling approximation, we find: 



In Tr,nax = N In M (12) 



The logarithm of the probability of the most probable distribution (of num- 

 bered balls in numbered baskets, or molecules in equal cells of coordinate- 

 space) is equal to the logarithm of the number of baskets (cells), multiplied 

 by the number of balls (molecules). 



Next suppose the caller, in a fit of uncontrollable zest for the game, calling 

 in succession every one of the conceivable distributions. What is the total 

 number of inventories compatible with all of them together? To sum over 

 every conceivable expression of the type of (1) seems a hopeless assignment, 

 but there is a short-cut to the result. 



Fix a particular order for the drawing of the balls from the sacks — it may 

 as well be the very order of their numbering. The iirst of the balls to be 

 drawn may be tossed into any one of the baskets, giving M distinct "possibil- 

 ities". The second may be tossed into any one of the baskets, the same or 

 another, giving in conjunction with the fate of the first M" different possibil- 

 ities. The third may be tossed — but we leap to the conclusion. There 

 are M-^' possibilities altogether, and these are the inventories. Thus the total 

 number of inventories consistent with all of the distributions, which I will 

 call TFtot , is a number whereof the logarithm is, 



In TFtot = iV In M (13) 



But this is the same as the expression for In TFmax ! 



The meaning of this strange coincidence can only be, that when N and 

 N/M are both so great that the super-Stirling approximation is a good one, 

 then the logarithm of the number of inventories belonging to the most prob- 

 able distribution is nearly as great as the logarithm of the total number of 

 inventories belonging to all of the distributions put together — so nearly as 

 great, that either logarithm is a good approximation to the other. 



In the foregoing very important paragraph, I have italicized the word 

 "logarithm" because if it were left out the statement would become a false 

 one. The statement is not true if applied to the numbers themselves. IFtot 

 is manyfold greater than TFmax, and the ratio between the two actually in- 

 creases with rising N . So does the difference between In TT'tot and In TFmax 

 increase with rising .V, but not so fast as either by itself; wherefore the truth 



