16 Henry Quastler 



instance, it was possible to approximate optimal efficiency by symbolizing 

 groups of events instead of single events. The same principle works in the case 

 of probabilities which are not integral powers of (1/2). We will illustrate 

 the method in the case of a situation with two alternatives. 



Example: Let there be two categories of events, 'A' and 'B', with associated 

 probabilities, p{K) and /?(B) : 



/'(A) = .7 



/KB) = .3 



The limiting value of symbols per event is: 



-IKO log2 AO = -(0.7 logo 0.7 + 0.3 log2 0.3) = 0.881291 . . . 



i 



If this situation is to be represented on the basis of single events, then one 

 needs one binary digit per event. 



Event Probability Representation 



A 0.7 1 



B 0.3 



Average number: 1.0 symbol per event; excess 12 per cent. 



The following two-event clusters are possible: AA, AB, BA, BB. If the two 

 events are independent, then the probability that both occur is the product 

 of their individual probabilities : 



p(AA) =7;(A) -piA), p(BA) =/7(B) - p{A\ etc. 



Setting up a Fano code, we get: 



Event Probability Representation 



AA .49 



AB .21 



BA .21 



BB .09 



Average 1.81, or 0.905 symbols per event; excess 3 per cent. 



If we can encode groups of three real events, then we get still closer to optimum 

 economy : 



Event Probability Representation 



AAA .343 



AAB .147 



ABA .147 



BAA .147 



ABB .063 



BAB .063 



BBA .063 



BBB .027 



Average: 2.686, or 0.895 digit per event; excess 1^ per cent. 



