Reprinted with corrections from The Bell System Technical Journal

p)and the digram probabilities p(i,)are related by the following formulas: p(=∑p,)=∑pU,)=∑pU)p① p(i,)=p(i)pU） ∑p)=∑p)=∑p,)=1. As a specific example suppose there are three letters A,B,C with the probability tables: Pi(i) i p() p6,) j AB A B A 0 A 品 A 0 4 方 i B 112 0 3 i B 多 0 品 C 子 贪 A typical message from this source is the following: ABBABABABABABABBBABBBBBABABABABABBBACACAB BABBBBABBABACBBBABA. The next increase in complexity would involve trigram frequencies but no more.The choice of a letter would depend on the preceding two letters but not on the message before that point.A set of trigram frequencies p(i,j,k)or equivalently a set of transition probabilities p(k)would be required.Continuing in this way one obtains successively more complicated stochastic pro- cesses.In the general n-gram case a set of n-gram probabilities p(i,i2,...,in)or of transition probabilities p(in)is required to specify the statistical structure. (D)Stochastic processes can also be defined which produce a text consisting of a sequence of “words.”Suppose there are five letters A,B,C,D,E and 16“words”in the language with associated probabilities: .10A .16 BEBE .11 CABED 04 DEB 04 ADEB .04 BED 05 CEED .15 DEED 05 ADEE 02 BEED .08 DAB .01 EAB 01 BADD05 CA .04 DAD .05EE Suppose successive "words"are chosen independently and are separated by a space.A typical message might be: DAB EE A BEBE DEED DEB ADEE ADEE EE DEB BEBEBEBE BEBE ADEE BED DEED DEED CEED ADEE A DEEDDEED BEBE CABED BEBE BED DAB DEED ADEB. If all the words are of finite length this process is equivalent to one of the preceding type,but the description may be simpler in terms of the word structure and probabilities.We may also generalize here and introduce transition probabilities between words,etc. These artificial languages are useful in constructing simple problems and examples to illustrate vari- ous possibilities.We can also approximate to a natural language by means of a series of simple artificial languages.The zero-order approximation is obtained by choosing all letters with the same probability and independently.The first-order approximation is obtained by choosing successive letters independently but each letter having the same probability that it has in the natural language.5 Thus,in the first-order ap- proximation to English,E is chosen with probability.12(its frequency in normal English)and W with probability.02,but there is no influence between adjacent letters and no tendency to form the preferred 5Letter,digram and trigram frequencies are given in Secret and Urgent by Fletcher Pratt,Blue Ribbon Books,1939.Word frequen- cies are tabulated in Relative Frequency of English Speech Sounds.G.Dewey,Harvard University Press,1923. 6

pi( j) and the digram probabilities p(i; j) are related by the following formulas: p(i) = ∑ j p(i; j) = ∑ j p( j; i) = ∑ j p( j) pj (i) p(i; j) = p(i) pi( j) ∑ j pi( j) = ∑ i p(i) = ∑ i; j p(i; j) = 1: As a specific example suppose there are three letters A, B, C with the probability tables: pi( j) j AB C A 0 4 5 1 5 i B 1 2 1 2 0 C 1 2 2 5 1 10 i p(i) A 9 27 B 16 27 C 2 27 p(i; j) j AB C A 0 4 15 1 15 i B 8 27 8 27 0 C 1 27 4 135 1 135 A typical message from this source is the following: ABBABABABABABABBBABBBBBABABABABABBBACACAB B A B B B B A B B A B A C B B B A B A. The next increase in complexity would involve trigram frequencies but no more. The choice of a letter would depend on the preceding two letters but not on the message before that point. A set of trigram frequencies p(i; j; k) or equivalently a set of transition probabilities pi j (k) would be required. Continuing in this way one obtains successively more complicated stochastic pro￾cesses. In the general n-gram case a set of n-gram probabilities p(i1 ; i2 ;:::; in) or of transition probabilities pi1 ;i2 ;:::;in￾1 (in) is required to specify the statistical structure. (D) Stochastic processes can also be defined which produce a text consisting of a sequence of “words.” Suppose there are five letters A, B, C, D, E and 16 “words” in the language with associated probabilities: .10 A .16 BEBE .11 CABED .04 DEB .04 ADEB .04 BED .05 CEED .15 DEED .05 ADEE .02 BEED .08 DAB .01 EAB .01 BADD .05 CA .04 DAD .05 EE Suppose successive “words” are chosen independently and are separated by a space. A typical message might be: DAB EE A BEBE DEED DEB ADEE ADEE EE DEB BEBE BEBE BEBE ADEE BED DEED DEED CEED ADEE A DEED DEED BEBE CABED BEBE BED DAB DEED ADEB. If all the words are of finite length this process is equivalent to one of the preceding type, but the description may be simpler in terms of the word structure and probabilities. We may also generalize here and introduce transition probabilities between words, etc. These artificial languages are useful in constructing simple problems and examples to illustrate vari￾ous possibilities. We can also approximate to a natural language by means of a series of simple artificial languages. The zero-order approximation is obtained by choosing all letters with the same probability and independently. The first-order approximation is obtained by choosing successive letters independently but each letter having the same probability that it has in the natural language.5 Thus, in the first-order ap￾proximation to English, E is chosen with probability .12 (its frequency in normal English) and W with probability .02, but there is no influence between adjacent letters and no tendency to form the preferred 5Letter, digram and trigram frequencies are given in Secret and Urgent by Fletcher Pratt, Blue Ribbon Books, 1939. Word frequen￾cies are tabulated in Relative Frequency of English Speech Sounds, G. Dewey, Harvard University Press, 1923. 6

digrams such as TH,ED,etc.In the second-order approximation,digram structure is introduced.After a letter is chosen,the next one is chosen in accordance with the frequencies with which the various letters follow the first one.This requires a table of digram frequencies p().In the third-order approximation, trigram structure is introduced.Each letter is chosen with probabilities which depend on the preceding two letters. 3.THE SERIES OF APPROXIMATIONS TO ENGLISH To give a visual idea of how this series of processes approaches a language,typical sequences in the approx- imations to English have been constructed and are given below.In all cases we have assumed a 27-symbol "alphabet,"the 26 letters and a space. 1.Zero-order approximation(symbols independent and equiprobable). XFOML RXKHRJFFJUJ ZLPWCFWKCYJ FFJEYVKCOSGHYD OPAAMKBZAACIBZL- HJOD. 2.First-order approximation(symbols independent but with frequencies of English text). OCRO HLI RGWR NMIELWIS EU LL NBNESEBYA TH EEI ALHENHTTPA OOBTTVA NAH BRL. 3.Second-order approximation(digram structure as in English). ON IE ANTSOUTINYS ARE T INCTORE ST BE S DEAMY ACHIN D ILONASIVE TU- COOWE AT TEASONARE FUSO TIZIN ANDY TOBE SEACE CTISBE. 4.Third-order approximation(trigram structure as in English). IN NO IST LAT WHEY CRATICT FROURE BIRS GROCID PONDENOME OF DEMONS- TURES OF THE REPTAGIN IS REGOACTIONA OF CRE 5.First-order word approximation.Rather than continue with tetragram,...,n-gram structure it is easier and better to jump at this point to word units.Here words are chosen independently but with their appropriate frequencies. REPRESENTING AND SPEEDILY IS AN GOOD APT OR COME CAN DIFFERENT NAT- URAL HERE HE THE A IN CAME THE TO OF TO EXPERT GRAY COME TO FURNISHES THE LINE MESSAGE HAD BE THESE. 6.Second-order word approximation.The word transition probabilities are correct but no further struc- ture is included. THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH WRITER THAT THE CHAR- ACTER OF THIS POINTIS THEREFORE ANOTHER METHOD FOR THE LETTERS THAT THE TIME OF WHO EVER TOLD THE PROBLEM FOR AN UNEXPECTED The resemblance to ordinary English text increases quite noticeably at each of the above steps.Note that these samples have reasonably good structure out to about twice the range that is taken into account in their construction.Thus in(3)the statistical process insures reasonable text for two-letter sequences,but four- letter sequences from the sample can usually be fitted into good sentences.In(6)sequences of four or more words can easily be placed in sentences without unusual or strained constructions.The particular sequence of ten words"attack on an English writer that the character of this"is not at all unreasonable.It appears then that a sufficiently complex stochastic process will give a satisfactory representation of a discrete source. The first two samples were constructed by the use of a book of random numbers in conjunction with (for example 2)a table of letter frequencies.This method might have been continued for(3),(4)and(5), since digram,trigram and word frequency tables are available,but a simpler equivalent method was used. 7

digrams such as TH, ED, etc. In the second-order approximation, digram structure is introduced. After a letter is chosen, the next one is chosen in accordance with the frequencies with which the various letters follow the first one. This requires a table of digram frequencies pi( j). In the third-order approximation, trigram structure is introduced. Each letter is chosen with probabilities which depend on the preceding two letters. 3. THE SERIES OF APPROXIMATIONS TO ENGLISH To give a visual idea of how this series of processes approaches a language, typical sequences in the approx￾imations to English have been constructed and are given below. In all cases we have assumed a 27-symbol “alphabet,” the 26 letters and a space. 1. Zero-order approximation (symbols independent and equiprobable). XFOML RXKHRJFFJUJ ZLPWCFWKCYJ FFJEYVKCQSGHYD QPAAMKBZAACIBZL￾HJQD. 2. First-order approximation (symbols independent but with frequencies of English text). OCRO HLI RGWR NMIELWIS EU LL NBNESEBYA TH EEI ALHENHTTPA OOBTTVA NAH BRL. 3. Second-order approximation (digram structure as in English). ON IE ANTSOUTINYS ARE T INCTORE ST BE S DEAMY ACHIN D ILONASIVE TU￾COOWE AT TEASONARE FUSO TIZIN ANDY TOBE SEACE CTISBE. 4. Third-order approximation (trigram structure as in English). IN NO IST LAT WHEY CRATICT FROURE BIRS GROCID PONDENOME OF DEMONS￾TURES OF THE REPTAGIN IS REGOACTIONA OF CRE. 5. First-order word approximation. Rather than continue with tetragram, ::: , n-gram structure it is easier and better to jump at this point to word units. Here words are chosen independently but with their appropriate frequencies. REPRESENTING AND SPEEDILY IS AN GOOD APT OR COME CAN DIFFERENT NAT￾URAL HERE HE THE A IN CAME THE TO OF TO EXPERT GRAY COME TO FURNISHES THE LINE MESSAGE HAD BE THESE. 6. Second-order word approximation. The word transition probabilities are correct but no further struc￾ture is included. THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH WRITER THAT THE CHAR￾ACTER OF THIS POINT IS THEREFORE ANOTHER METHOD FOR THE LETTERS THAT THE TIME OF WHO EVER TOLD THE PROBLEM FOR AN UNEXPECTED. The resemblance to ordinary English text increases quite noticeably at each of the above steps. Note that these samples have reasonably good structure out to about twice the range that is taken into account in their construction. Thus in (3) the statistical process insures reasonable text for two-letter sequences, but four￾letter sequences from the sample can usually be fitted into good sentences. In (6) sequences of four or more words can easily be placed in sentences without unusual or strained constructions. The particular sequence of ten words “attack on an English writer that the character of this” is not at all unreasonable. It appears then that a sufficiently complex stochastic process will give a satisfactory representation of a discrete source. The first two samples were constructed by the use of a book of random numbers in conjunction with (for example 2) a table of letter frequencies. This method might have been continued for (3), (4) and (5), since digram, trigram and word frequency tables are available, but a simpler equivalent method was used. 7

To construct(3)for example,one opens a book at random and selects a letter at random on the page.This letter is recorded.The book is then opened to another page and one reads until this letter is encountered. The succeeding letter is then recorded.Turning to another page this second letter is searched for and the succeeding letter recorded,etc.A similar process was used for(4),(5)and(6).It would be interesting if further approximations could be constructed,but the labor involved becomes enormous at the next stage. 4.GRAPHICAL REPRESENTATION OF A MARKOFF PROCESS Stochastic processes of the type described above are known mathematically as discrete Markoff processes and have been extensively studied in the literature.6 The general case can be described as follows:There exist a finite number of possible"states"of a system;S1,S2,...,Sn.In addition there is a set of transition probabilities;pi)the probability that if the system is in state Si it will next go to state S.To make this Markoff process into an information source we need only assume that a letter is produced for each transition from one state to another.The states will correspond to the"residue of influence"from preceding letters. The situation can be represented graphically as shown in Figs.3,4 and 5.The"states"are the junction Fig.3-A graph corresponding to the source in example B. points in the graph and the probabilities and letters produced for a transition are given beside the correspond- ing line.Figure 3 is for the example B in Section 2,while Fig.4 corresponds to the example C.In Fig.3 B Fig.4-A graph corresponding to the source in example C. there is only one state since successive letters are independent.In Fig.4 there are as many states as letters. If a trigram example were constructed there would be at most nstates corresponding to the possible pairs of letters preceding the one being chosen.Figure 5 is a graph for the case of word structure in example D. Here S corresponds to the“space'”symbol. 5.ERGODIC AND MIXED SOURCES As we have indicated above a discrete source for our purposes can be considered to be represented by a Markoff process.Among the possible discrete Markoff processes there is a group with special properties of significance in communication theory.This special class consists of the "ergodic"processes and we shall call the corresponding sources ergodic sources.Although a rigorous definition of an ergodic process is somewhat involved,the general idea is simple.In an ergodic process every sequence produced by the process 6For a detailed treatment see M.Frechet,Methode des fonctions arbitraires.Theorie des evenements en chaine dans le cas d'un nombre fini d'etats possibles.Paris,Gauthier-Villars,1938 8

To construct (3) for example, one opens a book at random and selects a letter at random on the page. This letter is recorded. The book is then opened to another page and one reads until this letter is encountered. The succeeding letter is then recorded. Turning to another page this second letter is searched for and the succeeding letter recorded, etc. A similar process was used for (4), (5) and (6). It would be interesting if further approximations could be constructed, but the labor involved becomes enormous at the next stage. 4. GRAPHICAL REPRESENTATION OF A MARKOFF PROCESS Stochastic processes of the type described above are known mathematically as discrete Markoff processes and have been extensively studied in the literature.6 The general case can be described as follows: There exist a finite number of possible “states” of a system; S1 ; S2;:::; Sn. In addition there is a set of transition probabilities; pi( j) the probability that if the system is in state Si it will next go to state Sj. To make this Markoff process into an information source we need only assume that a letter is produced for each transition from one state to another. The states will correspond to the “residue of influence” from preceding letters. The situation can be represented graphically as shown in Figs. 3, 4 and 5. The “states” are the junction A B C D E .1 .1 .2 .2 .4 Fig. 3—A graph corresponding to the source in example B. points in the graph and the probabilities and letters produced for a transition are given beside the correspond￾ing line. Figure 3 is for the example B in Section 2, while Fig. 4 corresponds to the example C. In Fig. 3 A A B B C B C .1 .5 .5 .5 .2 .8 .4 Fig. 4—A graph corresponding to the source in example C. there is only one state since successive letters are independent. In Fig. 4 there are as many states as letters. If a trigram example were constructed there would be at most n2 states corresponding to the possible pairs of letters preceding the one being chosen. Figure 5 is a graph for the case of word structure in example D. Here S corresponds to the “space” symbol. 5. ERGODIC AND MIXED SOURCES As we have indicated above a discrete source for our purposes can be considered to be represented by a Markoff process. Among the possible discrete Markoff processes there is a group with special properties of significance in communication theory. This special class consists of the “ergodic” processes and we shall call the corresponding sources ergodic sources. Although a rigorous definition of an ergodic process is somewhat involved, the general idea is simple. In an ergodic process every sequence produced by the process 6For a detailed treatment see M. Fr´echet, M´ethode des fonctions arbitraires. Th´eorie des ´ev´enements en chaˆıne dans le cas d’un nombre fini d’´etats possibles. Paris, Gauthier-Villars, 1938. 8

is the same in statistical properties.Thus the letter frequencies,digram frequencies,etc.,obtained from particular sequences,will,as the lengths of the sequences increase,approach definite limits independent of the particular sequence.Actually this is not true of every sequence but the set for which it is false has probability zero.Roughly the ergodic property means statistical homogeneity. All the examples of artificial languages given above are ergodic.This property is related to the structure of the corresponding graph.If the graph has the following two properties?the corresponding process will be ergodic: 1.The graph does not consist of two isolated parts A and B such that it is impossible to go from junction points in part A to junction points in part B along lines of the graph in the direction of arrows and also impossible to go from junctions in part B to junctions in part A. 2.A closed series of lines in the graph with all arrows on the lines pointing in the same orientation will be called a"circuit."The"length"of a circuit is the number of lines in it.Thus in Fig.5 series BEBES is a circuit of length 5.The second property required is that the greatest common divisor of the lengths of all circuits in the graph be one. Fig.5-A graph corresponding to the source in example D. If the first condition is satisfied but the second one violated by having the greatest common divisor equal to d>1,the sequences have a certain type of periodic structure.The various sequences fall into d different classes which are statistically the same apart from a shift of the origin (i.e.,which letter in the sequence is called letter 1).By a shift of from 0 up to d-I any sequence can be made statistically equivalent to any other.A simple example with d=2 is the following:There are three possible letters a,b,c.Letter a is followed with either b or c with probabilities and respectively.Either b or c is always followed by letter a.Thus a typical sequence is abacacacabacaba b aca c. This type of situation is not of much importance for our work. If the first condition is violated the graph may be separated into a set of subgraphs each of which satisfies the first condition.We will assume that the second condition is also satisfied for each subgraph.We have in this case what may be called a"mixed"source made up of a number of pure components.The components correspond to the various subgraphs.If LI,L2,L3,...are the component sources we may write L=p1L1+p2L2+p3L3+… 7These are restatements in terms of the graph of conditions given in Frechet. 9

is the same in statistical properties. Thus the letter frequencies, digram frequencies, etc., obtained from particular sequences, will, as the lengths of the sequences increase, approach definite limits independent of the particular sequence. Actually this is not true of every sequence but the set for which it is false has probability zero. Roughly the ergodic property means statistical homogeneity. All the examples of artificial languages given above are ergodic. This property is related to the structure of the corresponding graph. If the graph has the following two properties7 the corresponding process will be ergodic: 1. The graph does not consist of two isolated parts A and B such that it is impossible to go from junction points in part A to junction points in part B along lines of the graph in the direction of arrows and also impossible to go from junctions in part B to junctions in part A. 2. A closed series of lines in the graph with all arrows on the lines pointing in the same orientation will be called a “circuit.” The “length” of a circuit is the number of lines in it. Thus in Fig. 5 series BEBES is a circuit of length 5. The second property required is that the greatest common divisor of the lengths of all circuits in the graph be one. S S S A A A A A B B B B B B B C D D D D D D E E E E E E E E E E E Fig. 5—A graph corresponding to the source in example D. If the first condition is satisfied but the second one violated by having the greatest common divisor equal to d > 1, the sequences have a certain type of periodic structure. The various sequences fall into d different classes which are statistically the same apart from a shift of the origin (i.e., which letter in the sequence is called letter 1). By a shift of from 0 up to d ￾ 1 any sequence can be made statistically equivalent to any other. A simple example with d = 2 is the following: There are three possible letters a; b; c. Letter a is followed with either b or c with probabilities 1 3 and 2 3 respectively. Either b or c is always followed by letter a. Thus a typical sequence is abacacacabacababacac: This type of situation is not of much importance for our work. If the first condition is violated the graph may be separated into a set of subgraphs each of which satisfies the first condition. We will assume that the second condition is also satisfied for each subgraph. We have in this case what may be called a “mixed” source made up of a number of pure components. The components correspond to the various subgraphs. If L1, L2, L3 ;::: are the component sources we may write L = p1L1 + p2L2 + p3L3 +


7These are restatements in terms of the graph of conditions given in Fr´echet. 9

where pi is the probability of the component source Li Physically the situation represented is this:There are several different sources LI,L2,L3,...which are each of homogeneous statistical structure(i.e.,they are ergodic).We do not know a priori which is to be used,but once the sequence starts in a given pure component Li,it continues indefinitely according to the statistical structure of that component. As an example one may take two of the processes defined above and assume pi=.2 and p2=.8.A sequence from the mixed source L=.2L1+.8L2 would be obtained by choosing first LI or L2 with probabilities.2 and.8 and after this choice generating a sequence from whichever was chosen. Except when the contrary is stated we shall assume a source to be ergodic.This assumption enables one to identify averages along a sequence with averages over the ensemble of possible sequences(the probability of a discrepancy being zero).For example the relative frequency of the letter A in a particular infinite sequence will be,with probability one,equal to its relative frequency in the ensemble of sequences. If P is the probability of state i and p)the transition probability to state j,then for the process to be stationary it is clear that the P must satisfy equilibrium conditions: P=∑PpU) In the ergodic case it can be shown that with any starting conditions the probabilities P,(N)of being in state j after N symbols,approach the equilibrium values as N. 6.CHOICE,UNCERTAINTY AND ENTROPY We have represented a discrete information source as a Markoff process.Can we define a quantity which will measure,in some sense,how much information is"produced"by such a process,or better,at what rate information is produced? Suppose we have a set of possible events whose probabilities of occurrence are pi,p2,...,Pn.These probabilities are known but that is all we know concerning which event will occur.Can we find a measure of how much"choice"is involved in the selection of the event or of how uncertain we are of the outcome? Ifthere is such a measure,say H(p,p2,...,p),it is reasonable to require of it the following properties: 1.H should be continuous in the pi. 2.If all the pi are equal,p=,then H should be a monotonic increasing function of n.With equally likely events there is more choice,or uncertainty,when there are more possible events. 3.If a choice be broken down into two successive choices,the original H should be the weighted sum of the individual values of H.The meaning of this is illustrated in Fig.6.At the left we have three 1/2· .1/2 1/2 1/3 2/3.1/3 1/6. 1/2 1/3.1/6 Fig.6-Decomposition of a choice from three possibilities. possibilities p=,p2=,p3=.On the right we first choose between two possibilities each with probability and if the second occurs make another choice with probabilities,.The final results have the same probabilities as before.We require,in this special case,that H(,,若)=H(,)+H(层，) The coefficient is because this second choice only occurs half the time. 10

where pi is the probability of the component source Li. Physically the situation represented is this: There are several different sources L1, L2, L3 ;::: which are each of homogeneous statistical structure (i.e., they are ergodic). We do not know a priori which is to be used, but once the sequence starts in a given pure component Li, it continues indefinitely according to the statistical structure of that component. As an example one may take two of the processes defined above and assume p1 = :2 and p2 = :8. A sequence from the mixed source L = :2L1 + :8L2 would be obtained by choosing first L1 or L2 with probabilities .2 and .8 and after this choice generating a sequence from whichever was chosen. Except when the contrary is stated we shall assume a source to be ergodic. This assumption enables one to identify averages along a sequence with averages over the ensemble of possible sequences (the probability of a discrepancy being zero). For example the relative frequency of the letter A in a particular infinite sequence will be, with probability one, equal to its relative frequency in the ensemble of sequences. If Pi is the probability of state i and pi( j) the transition probability to state j, then for the process to be stationary it is clear that the Pi must satisfy equilibrium conditions: Pj = ∑ i Pipi( j): In the ergodic case it can be shown that with any starting conditions the probabilities Pj (N) of being in state j after N symbols, approach the equilibrium values as N ! ∞. 6. CHOICE, UNCERTAINTY AND ENTROPY We have represented a discrete information source as a Markoff process. Can we define a quantity which will measure, in some sense, how much information is “produced” by such a process, or better, at what rate information is produced? Suppose we have a set of possible events whose probabilities of occurrence are p1 ; p2 ;:::; pn. These probabilities are known but that is all we know concerning which event will occur. Can we find a measure of how much “choice” is involved in the selection of the event or of how uncertain we are of the outcome? If there is such a measure, say H ( p1 ; p2 ;:::; pn), it is reasonable to require of it the following properties: 1. H should be continuous in the pi. 2. If all the pi are equal, pi = 1 n , then H should be a monotonic increasing function of n. With equally likely events there is more choice, or uncertainty, when there are more possible events. 3. If a choice be broken down into two successive choices, the original H should be the weighted sum of the individual values of H. The meaning of this is illustrated in Fig. 6. At the left we have three 1/2 1/3 1/6 1/2 1/2 2/3 1/3 1/2 1/3 1/6 Fig. 6—Decomposition of a choice from three possibilities. possibilities p1 = 1 2 , p2 = 1 3 , p3 = 1 6 . On the right we first choose between two possibilities each with probability 1 2 , and if the second occurs make another choice with probabilities 2 3 , 1 3 . The final results have the same probabilities as before. We require, in this special case, that H ( 1 2 ; 1 3 ; 1 6 ) = H ( 1 2 ; 1 2 ) + 1 2H ( 2 3 ; 1 3 ): The coefficient 1 2 is because this second choice only occurs half the time. 10