QUANTITY OF VARIETY 7/23 



It is not necessary that the transformation should be closed. 

 Thus if the same set of ten letters is transformed by Y: 



P il P 



giving q q p p p p p p q p, the variety falls. It is easy to see that 

 only when the transformation is one-one (over the letters that 

 actually occur in the set to be transformed) is the set's variety 

 unchanged; and this is a very special type of transformation. 



Ex. 1 : Write the letters ^ to Z in a row; under it, letter by letter, write the first 

 26 letters of some well known phrase. The transition from upper row to 

 lower now defines a single-valued transformation (u). Write your name 

 in full, find the variety among its letters, transform by // (i.e. "code" it) 

 and find the variety in the new set of letters How has the variety changed ? 

 Apply u repeatedly; draw a graph of how the variety changes step by step. 



Ex. 2: In a certain genus of parasite, each species feeds off" only one species of 

 host. If the varieties (in our sense) of parasites' species and hosts' species 

 are unequal, which is the larger? 



Ex. 3: "A multiplicity of genotypes may show the same phenotypic feature." 

 If the change from each genotype to its corresponding phenotype is a 

 transformation V, what change in variety does F cause? 



Ex. 4: When a tea-taster tastes a cup of tea, he can be regarded as responsible 

 for a transformation Kconverting "sample of leaf" as operand to "opinion" 

 as transform. If the taster is perfect, Y will be one-one. How would he 

 be described if Y were many-one ? 



Ex. 5 : When read to the nearest degree on each of seven occasions, the tempera- 

 tures of the room and of a thermostatically-controlled water-bath were 

 found to be 



Room: 65, 62, 68, 63, 62, 59, 61. 

 Water-bath: 97, 97, 98, 97, 97, 97, 97. 



How much variety is shown (i) by the room's temperatures, (ii) by those 

 of the bath? What would have been said had the variety in (i) 

 exceeded that of (ii)? 



*£'.v. 6: If the transformation has been formed by letting each state go to one 

 state selected at random from all the states (independently and with equal 

 probabilities), show that if the number of states is large, the variety will 

 fall at the first step, in the ratio of 1 to 1 — Xje, i.e. to about two-thirds. 

 (Hint: The problem is equivalent (for a single step) to the following: n 

 hunters come suddenly on a herd of n deer. Each fires one shot at a deer 

 chosen at random. Each bullet kills one and only one animal. How many 

 deer will, on the average, be hit? And to what does the average tend as 

 // tends to infinity?) 



7/23. Set and machine. We must now be clear about how a set 

 of states can be associated with a machine, for no real machine 

 can, at one time, be in more than one state. A set of states can be 

 considered for several reasons. 



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