54 



In this case for each table separately — if we still distinguish between 

 the three different holes ') — the most probable division of the balls is : 

 two red ones and one black one. For this the probability is') 



2 2 13/ 12 



3 ^ 3 3 2/ 27* 



We must now ask: what is the most probable distribution of the 

 combinations over all the 27 tables? We can here still distinguish 

 between the tables. As the most probable distribution we get that 

 in which on only twelve tables two red balls and one black ball 

 lie, on eight of the others three red ones, on six 2 black ones and 

 one red one, and on the last one three black balls. For this 

 distribution the probability is expressed by 



'""~V27J * V27J ■ V27J V27/ " 67^127* 



On the other hand, the chance that on each of the 27 tables 

 uniformly two red balls and one black ball should come to lie is given by 



/12' 

 V27 

 The ratio between the two is 



Wu 1227-12. 6.' 8 7 12/ 



W„, 88 . 66 . 27./ 



which is very much smaller than 1 '). 



Let us now suppose the number of balls that can lie on a table, 

 and also the number of tables to be greater; the number of different 

 typical possibilities of division on each table separately (varying from 

 all red to all black) then rises, as also the number of ways in which 

 we can find these types of division spread over the collective tables. 



The chance of the most probable division for one particular table 

 becomes smaller. The probability TF„, that just this division will be 

 found repeated on every table, becomes therefore represented by a 

 high power of a very small fraction. 



1) That is to say, if for a particular combination (e.g. 1 red, 2 black) we count 

 as separate possibilities the cases in which differently coloured balls lie in a given 

 hole. 



') The chance of all three being red is ^«y, of one red and two black ^, of 

 all three black ^. 



, ^ 6/8.' 12/12-^"-i2 6/ 8/ 12 12 12 



'^ ^''WW-^f:^ ^6^-8^-13"-T4 • • • • 27' ^"^h^^h further 



6/ 18/ 1 12 12 12 



66" ^ G4- P" ^ 4(70' Ï3 ^ ^' 14 "^ ^' ' " ■ 27 ^ '■ 



