ANSWERS TO THE EXERCISES 14/1 



1 

 5: 1-4 bits; more accurately it is (1 +-j- + . . .) log2 c. 6: Examina- 

 tion of the kth card in a pack of n gives information, or has entropy, 

 1 1 n— k n — k 



~ n- k + I '°^ « - A; + 1 ~ n- k + I ^^ n- k + I 



if the drawing occurs. If success has occurred earUer the entropy is 0. 

 These two events (and their entropies) have probabilities (n — k + 1)1 it 

 and {k — l)/n. So the weighted average entropy is 



1 / 1 n - k \ 



log 1——. + (n-k) log 1— — : 



n \ n — k + I n — k + 1/ 



which is -[(/? - k + l)log(n - k + I) -(n - k) log {n - Ar)|. 



7: At each drawing the entropy is the same — that of the probabilities 



1 // - 1 



— and , and the average information 



n n 



-(« log n - {n - 1) log {n — 1)). 



14/1. 1: An adequate supplementary input of water is, of course, necessary. 

 The output comes from this, through a tap, which is controlled by the 

 input. A possible method is to use piston or bellows so that the pressure 

 set up when 0, 1 or 2 ml/sec are forced through a narrow orifice will 

 move the tap to the appropriate position. 



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