920 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 



where X = ^ Xi . This may be shown directly from the convexity of 

 the logarithm. 



and 



Thus we put 



(1 + ^) = 2q 

 (1 - -f) = 2p 



G^max = 1 + P log p + g log g 



= R 



which is the rate of transmission as defined by Shannon. 



One might still argue that the gambler should bet all his money 

 (make ^ = 1) in order to maximize his expected win after N times. It 

 is surely true that if the game were to be stopped after N bets the answer 

 to this question would depend on the relative values (to the gambler) 

 of being broke or possessing a fortune. If we compare the fates of two 

 gamblers, however, playing a nonterminating game, the one which uses 

 the value € found above will, with probability one, eventually get ahead 

 and stay ahead of one using any other i. At any rate, we will assume 

 that the gambler will always bet so as to maximize G. 



THE GENERAL CASE 



Let us now consider the case in which the channel has several input 

 symbols, not necessarily equally likely, which represent the outcome of 

 chance events. We will use the following notation: 



p{s) the probability that the transmitted symbol is the s'th one. 

 p(r/s) the conditional probability^ that the received symbol is the 



r'th on the hypothesis that the transmitted symbol is the s'th 



one. 

 p(s, r) the joint probability of the s'th transmitted and r'th received 



symbol. 

 q{r) received symbol probability. 

 q(s/r) conditional probability of transmitted symbol on hypothesis 



of received symbol, 

 a, the odds paid on the occurrence of the s'th transmitted symbol, 



i.e., as is the number of dollars returned for a one-dollar bet 



(including that one dollar), 

 a(s/r) the fraction of the gambler's capital that he decides to bet on 



the occurrence of the s'th transmitted symbol after observing 



the r'th received symbol 



^ 



