9/2 AN INTRODUCTION TO CYBERNETICS 



particular new state, may go to some one of the possible states, the 

 selection of the particular state being made by some method or 

 process that gives each state a constant probability of being the 

 transform. It is the unchangingness of the probability that 

 provides the law or orderliness on which definite statements can 

 be based. 



Such a transformation would be the following: x' = x + a, 

 where the value of a is found by spinning a coin and using the rule 

 Head: a = 1 ; Tail: a = 0. Thus, if the initial value of .t is 4, and 

 the coin gives the sequence TTHHHTHTTH, the trajectory 

 will be 4, 4, 4, 5, 6, 7, 7, 8, 8, 8, 9. If the coin gives HTHHTTT 

 HTT, the trajectory will be 4, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8. Thus the 

 transformation and the initial state are not sufficient to define a 

 unique trajectory, as was the case in S.2/17; they define only a set of 

 trajectories. The definition given here is supplemented by instruc- 

 tions from the coin (compare S.4/19), so that a single trajectory is 

 arrived at. 



The transformation could be represented (uniformly with the 

 previously used representations) as: 



3 4 5 



3 4 4 5 5 6 



where the ^ means that from state 3 the system will change 



with probability ^ to state 3, 

 snd ,, ,, ,, ,, ,, 4. 



Such a transformation, and especially the set of trajectories that 

 it may produce, is called "stochastic", to distinguish it from the 

 single-valued and determinate. 



Such a representation soon becomes unmanageable if many 

 transitions are possible from each state. A more convenient, and 

 fundamentally suitable, method is that by matrix, similar to that of 

 S.2/10. A matrix is constructed by writing the possible operands 

 in a row across the top, and the possible transforms in a column 

 down the left side; then, at the intersection of column / with 

 row 7, is put the probability that the system, if at state /, will go to 

 state/ 



As example, consider the transformation just described. If the 

 system was at state 4, and if the coin has a probability ^ of giving 



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