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Pros and cons of probabilisticflow‐graphsH. BRENY aa Institut de mathématique , Université de l'Etat , Liège, BelgiumPublished online: 09 Jul 2006.

To cite this article: H. BRENY (1977) Pros and cons of probabilistic flow‐graphs,International Journal of Mathematical Education in Science and Technology, 8:1, 69-78, DOI:10.1080/0020739770080110

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INT. J. MATH. EDUC. SCI. TECHNOL., 1977, VOL. 8, NO. 1, 69-78

Pros and cons of probabilistic flow-graphs

by H. BRENYInstitut de mathématique,

Université de l'Etat, Liège, Belgium

(Received 18 December 1975; revision received 19 May 1976)

It is argued that the existence of the probabilistic abacus can be an in-centive to teach probability in a way that blunts the students' critical sense,by giving them the habit of accepting hypotheses without seriously questioningtheir validity. This is first illustrated on two examples and then commentedon from a more general point of view. The conclusion is that the choice of aflow-graph must be preceded by the analysis of a tree and its probabilityfunction.

1. The probabilistic abacusThose who are interested in the teaching of probability in schools regard

Engel's [1, 2] recent invention of the probabilistic abacus as a significant advance.This abacus is an algorithm to solve problems connected with probabilisticflow-graphs, or more explicitly to compute the mean time to absorption and theabsorption probabilities for absorbing Markov chains with a finite set of statesand stationary transition probabilities (the algorithm can also cope with regularchains). So far, no formal proof of its general validity has been published, butit has been used for many different problems without ever encountering a case offailure. One can, therefore, safely assume that the abacus is indeed a reliabletool. Anyone who has tried it knows how efficient and easy it is to use.

Yet, in spite of, or perhaps just by reason of, it being such a good tool, theprobabilistic abacus can be quite dangerous, as there is a tendency in the teach-ing of probability in schools to teach in an objectionable way that is all themore likely to be followed when using the abacus. Of course, competentprobabilists do not follow that way, even if they advocate using probabilisticflow-graphs and abacus, but there are many teachers who, though they are quitecompetent as teachers, are hardly competent in probability and therefore prone tofollow any easy way shown to them. Such teachers are quite likely to teach in away that blunts their pupils' critical sense and generates in them false, or at leastmisleading, ideas about random situations. It could be argued that, in mathe-matics, very few methods work smoothly in all cases and that the use of flow-graphs just follows the common rule. But there is here something worse: thepossibility of giving children a wrong habit by getting them used to a method thatis far less general than they most probably would think, because it looks generaland its lack of generality needs to be pointed out explicitly.

We shall first try to show this in two examples and then come to a more generalconclusion. First, however, we must point to a danger that is not the least realfor involving inexperienced teachers (of probability). Once the flow-graph of arandom situation is established, ' playing ' the abacus has absolutely nothingrandom. It is by no means inconceivable that some teachers would so use theabacus that the very idea of randomness would be lost on their pupils: this would

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70 H. Breny

be serious indeed. But it must be admitted that, in such a case, the fault does notlie with the abacus (or the flow-graph method in general) but with the teacher—any tool, however good, can be misused.

2. First example: Ehrenfest-like urnsThe situation involves two (distinguishable) urns {A and B, say), a sequence

of'trials ', numbered 1, 2, . . . ,«, ... (with the ' epochs ' between them numbered0, 1, 2, ..., n, ...), and a random mechanism (to be further specified below) withtwo possible results only, G and R. At epoch 0, urn A contains one red (R)and two green (G) marbles, and urn B contains one green and two red ones. Ateach trial, the random mechanism chooses either G or R, and a marble is removedfrom one of the urns and put into the other; it has the colour (G or R) chosen bythe random mechanism and is taken from urn A at odd-numbered trials, from urnB at even-numbered ones; the game is over as soon as both urns are homogeneous(RRR or GGG). Here is a realized example:

Epochs 0 1 2 3 4 5 6Trials 1 2 3 4 5 6Random choices G G R R R GStates of urns GGR GR GGR GG GGR GG GGG

GRR GGRR GRR GRRR GRR GRRR RRR

Figure 1.

This is enough to describe Q, the ' set of possible outcomes ' that is the basicelement of the mathematical model of the situation. It can be visualized as theset of paths along the tree displayed in figure 1 (where a full branch means G,and a stippled one R); it is, of course, an infinite (but recurring) tree. The' states of the system ', are, obviously:

GGGRRR

GGGRRR

GGRGRR

GRGGRR

GRRGGR

RRGGGR

RRRGGG

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Pros and cons of probabilistic flow-graphs 71

Figure 2.

Figure 3.

Since every element in Q is a (finite or infinite) sequence with values in the aboveset of states, one might as well display D. as the set of paths through the ' pro-gressive diagram ' of figure 2. In the recent literature devoted to probabilityteaching [3-6] the advocated display of Q. would be the set of paths through theflow-graph of figure 3. A teacher conversant with that literature but little versedin probability theory proper might very well think that this display is the only oneworth using.

As displays of Q, the sets of paths through the tree of figure 1, the progressivediagram of figure 2, or the flow-graph of figure 3 are all equally good. None ofthese representations of D. makes any appeal to the properties of the randommechanism involved in the situation, which it should, for those propertiesdetermine the probability-function, to which O is logically anterior (proof:this function is defined on a subset of ^O) .

If the random mechanism is supposed f to be absolutely constant from trialto trial, i.e.

Vw; Pr(G at trial n) =pG & Pr(i? at trial n) =pR

then the events associated with the tree of figure 1 are probabilized by the relation

= an} = f[ pa.i

On the tree of figure 1, the diagram of figure 2, or the flow-graph of figure 3, thisis adequately represented by affixing numbers pa or pR, as the case may be, toevery branch or arrow. Thus, in that case, all three figures are perfectly adequaterepresentations of both D and Pr (figure 3 is, of course, the simplest).

fThe hypothetic character of such an assertion is obvious; in classroom situations,however, the random mechanism may be, and usually is, perfectly known (and can be usedin simulations).

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72 H. Breny

Now let us suppose that the random mechanism has a built-in periodiccomponent; e.g.

Vw ; Pr(G at trial n) = § if n =0 mod 3= \ if n s i mod 3= f i fws2mod3

Then, the branches of the tree in figure 1 have to bear numbers § and \, \ and| , f and i in successive columns. This is transferred without ado to the pro-gressive diagram of figure 2, but it cannot be transferred to the flow-graph offigure 3.

Finally, let us suppose that the random mechanism is hereditary:

(1) at trial 1, both G and R have probability \ ;

(2) from trial 2 on, two different sub-mechanisms are working, depending onthe realized result of the preceding trial;

if trial n — 1 has resulted in G, thenPr(G at trial « )=§

if trial n — 1 has resulted in R, thenPr(G at trial n) = \

Then, the branches of the tree have to bear

(a) numbers \ for trial 1;

(b) numbers §, \, \, f, as the case may be, from trial 2 on.

This cannot be transferred to the progressive diagram nor, a fortiori, to theflow-graph. Indeed, there are states where the incoming arrows are one G andone R; the outgoing arrows would then have to carry two different numbers (asin figure 4). The example clearly shows that the privileged position given in thedidactic literature to the flow-graph display of Q. is unwarranted: contrary to theimpression given, it can be used in special cases only. (The random mechanismsmentioned above are all easily simulated in the classroom.)

Figure 4.

3. Second example: a rat in a mazeThe situation is now this: a rat is put into cell A of the four-cell maze depicted

in figure 5; it wanders through the various ' gates ' until it arrives at cell D,where it finds some food; the experience is then over. We suppose, of course,that the rat's behaviour has some random component (and, merely for ease ofanalytic representation, that the rat stays for ever in D once it has got there).

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Pros and cons of probabilistic flow-graphs

(food

73

1 BA 1A |

1 C

D ©

Figure 5.

The possible transitions are, of course,

A to B BtoD

AtoC BtoA

CtoD

Cto A DtoD

therefore, the set of possible outcomes, Q, can be described as: the set of allsequences with values A, B, C, D, whose first value is A and where every value Ais followed by either B or C, every value B or C by either A or D, and every valueD by D. Q is easily represented geometrically as the set of paths on a tree (figure6), or along a progressive diagram (figure 7) or a flow-graph (figure 8).

Figure 6.

epochstrials

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74 H. Breny

Let us suppose first that the rat is a very silly beast, without the slightest traceof intelligence or memory, and that the food pellets are completely odourless.Then, at each trial all possible transitions have the same probability, viz. \.Such a situation is adequately described by affixing numbers \ to every branchof the tree or every arrow of the progressive diagram or the flow-graph. Similarly,if the food pellets have a very faint odour, some transitions are more likely thanothers; one might have, for instance,

for both A->B and A-^C : \for both B->D and C->D : ffor both B^A and C-+A : i

Such a situation is again adequately described by affixing numbers to the arrowsor branches of the tree, the progressive diagram, or the flow-graph.

Not all rats are that silly, however. Here is a rat with some memory: whateverthe cell it happens to be in, it remembers which door it has entered it through, sothat it has always two choices open—either to progress, i.e. leave the present cellby the door it has not entered it through, or to retrogress, i.e. leave the presentcell by the same door as it has entered it. It also has a trace of intelligence, inthat it consistently prefers progressing to retrogressing:

at the start: Pr(A->B) = Vr{A-+C) = |for every other move (including those from cell A):

Pr(progression) = f, Pr(retrogression)= f

Moreover, it notices that the B-to-D door is slightly wider than all the others, sothat the 5 -KD transition probability (which is always that of a progression) has anadditional factor of § (and its complementary probability, B->A, an additionalfactor of T

7C).

Such a situation (again easily simulated in the classroomf) is translated into aprecise mathematical model by affixing numbers to the branches of the tree infigure 6: e.g.

1 7 3 2 3 1 82 S 5 S 5 2"5

But it cannot be described in that way on the progressive diagram: for instance,the arrow leaving cell A for cell B at trial 3 can as well be

a progression: or a retrogression:A->C->A->B A->B-*A-+B

I §

For the same reason, the situation cannot be depicted on the flow-graph.The conclusion here is similar to that in the previous paragraph.

fPupils could even investigate experimentally whether there is any advantage in beingintelligent (in the precise sense described in the text).

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Pros and cons of probabilistic flow-graphs 75

4. General commentsIt is generally accepted, nowadays, that the teaching of probability ought to

begin early. Of course, in the first stages, such a teaching is quite intuitive, veryinformal: its aim is to make the children familiar with the idea of randomness byhaving them contemplate, simulate, and analyse random situations. Situationslike those described in the preceding examples are quite likely to be met in schoolsand, if the trend of the present didactic literature is followed, analysed forthwithwith the aid of a flow-graph.

Now, this raises a serious point. It is agreed that the analysis is, at that stage,informal and it is therefore possible, even quite likely, that the very term ' set ofpossible outcomes ' (or its equivalent) will not even be mentioned; and, anyway,nobody denies that the flow-graph of a time-progressive situation is a gooddescription of that set (not the flow-graph itself, but the set of its paths). Thus,why object to the use of the flow-graph without the preliminary step of the tree ?And yet it is obvious that children for whom a time-progressive random situationis immediately translated into a flow-graph will not be able to think of any otherspecification of the probability than the continued multiplication of numbersaffixed to the arrows of the flow-graph—indeed, what else could they think of ?Worse than that, the teacher himself is quite likely to fall into the same trap; fora trap it is, as the examples analysed in §§ 2 and 3 (and one could exhibit a numberof similar examples) have clearly shown that: it is a wrong assumption that everytime-progressive random situationf can be probabilized by continued multi-plication along the arrows of its flow-graph.

It may be useful to restate the above in more general and abstract terms, inthe usual language of probability theory (if only for the benefit of teachers ac-quainted with that language, or probabilists interested in didactics).

// a ' system ' is capable of any of the states ex eg (whose set is denoted byK) and jumps at epochs 1, 2, ..., t, ... from one state to another (according tosome well-defined pattern of possible transitions), the jumps being made ac-cording to the ' results ' alt a2, ..., at of a given random mechanism R; thenthe mathematical model of the situation comprises :

(i) the set of possible outcomes, D.: the set of the sequences with values inK : Q = ^ N (notation : n-^wn)

(ii) the a-field of observable events, c¥ : the cr-field generated by the set 8$of ' begins ' :

Vr e N ; V(a0) ..., ar) eK*r+V;beg(a0, ..., ar)={w eQ^roo = ao& ... &cwr = ar)

(this means that begins are such elementary events that their observability isindisputable).

(iii) probability function, the one (uniquely) defined on c¥ by its values on 88,these last being inferred from the description of the random mechanism R (other-wise said: R must be well-enough defined for the assignment of a probability toeach begin to be possible and unambiguous). The theory of conditionalprobabilities shows that to each possible transition

(a0, ...,ary+ar+1

fEven with a finite set of possible states (the identification of the states is a good exercise,whatever the further analysis).

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76 H. Breny

is attached a well-defined number

p(a0, ..., a rK+1=Pr[beg(a0, ..., ar+1)\beg(a0, ..., ar)]and that

2 Piao, . . . , « r ) « = l

(distinct begins are either disjoint or nested). Conversely, according to thewell-known Daniell-Kolmogorov [7] theorem, any such assignment defines a(unique) probability on df. Since Q is a set of sequences, its most obviousdescription is through a progressive diagram (figure 9), every forward-goingbroken line through the nodes representing an element of Q. But this is not,in general, enough to support all the possible assignments of numbers p{u v)w.

epochs 0trials

/ \

\

1f

etc.

3Figure 9.

It is thus necessary to make use of the (much less easily depicted) tree with (inthe worst case) s branches issuing from each node. (The use of the tree isequivalent to the description of the sequence n^wn as n->(w0 wn).) Afortiori, the use of the directed graph (' flow-graph ') with vertices labelledeu ..., es and edges corresponding to the (possible) transitions ei-^ei is not, ingeneral, adequate to all possible assignments of numbers /><„,..._ v)u). Whenthe transition probabilities are markovian, i.e. when

Vn; Va'o, ..., a'n, a"0, ...,a"n; VM, V

P(a'o, .... an'u)v=p(a\ a\u)vthen and only then is it possible to use the progressive diagram to display boththe possible-outcomes set Q. and the function Pr. When they are markovian andstationary, i.e. when

Vn,^; Va'o, ..., a'n,a"0, ...,a"q; Vu,v

pWo a'nu)v=p(a"0> _ f a"qu)vthen and only then is it possible to use the flow-graph to that purpose.

The main aim of the teaching of probability in schools is to acquaint thechildren with the idea of randomness and the mathematical means of analysingrandom situations!. When time-progressive situations of the type just describedare analysed by the flow-graph method (a method that allows, in many cases,the building of an explicit solution—all the more because the probabilistic abacus

fA secondary aim is to acquaint them with the logical structure of a ' physical theorv '[8, 9].

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Pros and cons of probabilistic flow-graphs 77

can now be called to help), this aim is largely fulfilled. None the less, it is ful-filled at a price, and the price is heavy; for the systematic use of the method is inopposition to some of the general aims of mathematical education: instead ofgetting the habit of critically analysing a situation and making explicit the as-sumptions used in its mathematical model (or, preferably, models), the pupil isled to regard as self-evident that any probability can be defined by affixing numbersto the arrows of a flow-graph: and that is a great pity (all the more because thepupil is then led to mix the definitions of the possible-outcomes set and theprobability function; this will prove a severe handicap when he passes to a moredeductive teaching).

v 5. Modified flow-graphsIt is well known that a non-Markov chain can often be transformed into a

Markov one by enlarging the set of its states. This happens in both of ourexamples above (§§ 2 and 3). Indeed, in the case of the rat in a maze, with enoughmemory to discern progression from retrogression, all one has to do is to splitstate A into

state A as a starting point: (S)Astate A following state B: (B)Astate A following state C: (C)A;

then, flow-graphs hold their own again (figure 10). Similarly, in the case of theEhrenfest-like urns with the hereditary mechanism of § 2, all one has to do is tosplit the states GR/GGRR and GRRjGGR each into two, one arrived at by agreen arrow, the other by a red one (say, gGRjGGRR and rGRjGGRR, gGRRjGGR and rGRRjGGR) and the state GGR/GRR into three (start, gGGRjGRR,and rGGRjGRR) (figure 11). This trick works for practically all stationary(or time-periodic) random mechanisms that can be simulated in the classroom(because they all have a finite memory).

7/25 CB)A

1/2

C5)A

Figure 10.

Figure 11.

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78 Pros and cons of probabilistic flow-graphs

However, this fact does not avoid the objections raised in § 4. Rather, itpoints to the very important fact that a convenient flow-graph can be found onlyafter the fundamental tree has been probabilized. This is very different fromjumping straight to the plain flow-graph with (non-hereditary) arrow-probability.

Didactically, this fact justifies the following method of analysis. For anygiven time-progressive situation (with a finite set of states and a finite memory):

(1) identify the states of the system (this very good exercise is already partof the plain flow-graph method);

(2) display in the form of a tree the possible transitions of the system, makingsure that enough of the tree has been drawn to show all the possible casesat least once; /

(3) distribute the tree-vertices into classes, members of the same class havingidentical labels both for themselves (states) and for the adjacent branches(transition probabilities);

(4) draw a flow-graph with a ' micro-state ' for each of the classes identifiedat step 3;

(5) analyse this flow-graph in the usual way.Then, the objections raised in § 4 against the flow-graph method are all met, and,the method itself, with the attached tools (e.g. probabilistic abacus, generatingfunctions, method of additional event, stochastic matrices), far from beingbelittled, has its working range greatly extended; which is all to the good, for it isan excellent method indeed.

References[1] ENGEL, A., 1975, Educ. Stud. Math., 6, 1.[2] ENGEL, A., 1975, Mathematikunterricht, 21/2, 70.[3] RADE, L., 1973, Int. J. Math. Educ. Sci. Technol., 4, 363.[4] ENGEL, A., 1975, Mathematikunterricht, 21/2, 38.[5] ENGEL, A., 1973, Wahrscheinlichkeitsrechnung und Statistik. Bd. 1 (Stuttgart: Klett);

1974, Ibid., Bd. 2.[6] HOWARD, R. A., 1971, Dynamic Probabilistic Systems, Vols. 1, 2 (New York: Wiley).[7] BOURBAKI, N., 1969, Eléments de mathématique. Integration, Chap. 9 (Paris: Herman)

(notice historique, pp. 119-121).[8] BUNGE, M., 1973, Philosophy of Physics, Chaps. 3 and 4 (Dordrecht: Reidel).[9] BRENY, H. 1976, Enseign. math., XXII/1, 31.

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