His wife can sex them but knows nothing of breeds, how many distinct
Classes of chicks can they distinguish when working together?
Ex. 3: A spy in a house with four windows arranged rectangularly is to signal out
To sea at night by each window showing, or not showing, a light. How many
Forms can be shown if, in the darkness, the position of the lights relative to
The house cannot be perceived ?
Ex. 4: Bacteria of different species differ in their ability to metabolise various
Substances: thus lactose is destroyed by E. cold but not by E. typhi. If a bac-
Teriologist has available ten substances, each of which may be destroyed or
Not by a given species, what is the maximal number of species that he can
Distinguish ?
Ex. 5: If each Personality Test can distinguish five grades of its own character-
Istic, what is the least number of such tests necessary to distinguish the
Individuals of the world’s population?
Ex. 6: In a well-known card trick, the conjurer identifies a card thus: He shows
Cards to a by-stander, who selects, mentally, one of them without reveal-
Ing his choice. The conjurer then deals the 21 cards face upwards into three
Equal heaps, with the by-stander seeing the faces, and asks him to say which
Heap contains the selected card. He then takes up the cards, again deals them
Into three equal heaps, and again asks which heap contains the selected card,
And similarly for a third deal. The conjurer then names the selected card.
What variety is there in (i) the by-stander’s indications, (ii) the conjurer’s
Final selection ?
Ex. 7: (Continued.) 21 cards is not, in fact, the maximal number that could be
Used. What is the maximum, if the other conditions are unaltered?
Ex. 8: (Continued.) How many times would the by-stander have to indicate
Which of three heaps held the selected card if the conjurer were finally to be
Able to identify the correct card out of the full pack of 52?
Ex. 9: If a child’s blood group is O and its mother’s group is 0, how much variety
Is there in the groups of its possible fathers?
125
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
Q UA N TI TY O F V AR IE TY
It will have been noticed that many of the exercises involved
|
|
|
The finding of products and high powers. Such computations are
Often made easier by the use of logarithms. It is assumed that the
Reader is familiar with their basic properties, but one formula will
Be given for reference. If only logarithms to base a are available
And we want to find the logarithm to the base b of some number
N, then
Log aN
log bN = ---------------
Log ab
In particular, log2N = 3.322 log10N.
The word variety, in relation to a set of distinguishable ele-
Ments, will be used to mean either (i) the number of distinct ele-
Ments, or (ii) the logarithm to the base 2 of the number, the
Context indicating the sense used. When variety is measured in the
Logarithmic form its unit is the “bit”, a contraction of “BInary
DigiT”. Thus the variety of the sexes is 1 bit, and the variety of the
52 playing cards is 5.7 bits, because log2 52 = 3.322 log1052 =
3.322 x 1.7160 = 5.7. The chief advantage of this way of reckon-
Ing is that multiplicative combinations now combine by simple
Addition. Thus in Ex. 7/6/2 the farmer can distinguish a variety of
3 bits, his wife 1 bit, and the two together 3 + 1 bits, i.e. 4 bits.
To say that a set has “no” variety, that the elements are all of
One type, is, of course, to measure the variety logarithmically; for
The logarithm of 1 is 0.
Ex. 1: In Ex. 7/6/4 how much variety, in bits, does each substance distinguish?
Ex. 2: In Ex. 7/6/s: (i) how much variety in bits does each test distinguish? (ii)
What is the variety in bits of 2,000,000,000 distinguishable individuals?
From these two varieties check your previous answer.
Ex. 3: What is the variety in bits of the 26 letters of the alphabet?
Ex. 4: (Continued.) What is the variety, in bits, of a block of five letters (not
Restricted to forming a word) ? Check the answer by finding the number of
|
|
|
Such blocks, and then the variety.
Ex. 5: A question can be answered only by Yes or No; (i) what variety is in the
Answer? (ii) In twenty such answers made independently?
Ex. 6: (Continued.) How many objects can be distinguished by twenty questions,
Each of which can be answered only by Yes or No ?
Ex. 7: A closed and single-valued transformation is to be considered on six
States:
a b c d e f ↓ ? ? ? ? ? ?
Дата добавления: 2019-11-16; просмотров: 207; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!