CHANCE: a property attributed to unpredictable happenings


Amidst the infinite on-going chaos of the universe, some happenings are of sufficient interest to humans, that they are archived as 'events'. They are specific, physical, existential reconfigurations which have been described and labeled in time. 'Sunrise on 15 July 1944'...'World War One'...'the death of the last Moa'...'the flight of godwit E7 from Alaska to NZ in 2007'...'The Tunguska explosion on 30 June 1908'... 'the coin-toss for ends at the 2018 Australian Open tennis grand-slam mens singles final'...etc, are all verbal description-labels for archived information about these specific events.
'Chance' is a property that is attributed to some of those particular the Tunguska event or the coin-toss event... which appear to have been impossible to foretell.

Events vary in duration, complexity and pragmatic relevance, but any description of one will involve a few minimum considerations. It is not existentially possible for an event to be instantaneous, no matter how plausibly or persuasively it might be suggested. The archives of any event will need to record the fact that it had a 'beginning' and a 'duration' the very least...and an 'end' if some termination criteria has been fulfilled. Thus for example, from a grand and complex perspective, the 'Triassic' period was an event that lasted about 50 million years, had a 'beginning' about 250 million years ago, and an 'end' about 200million years ago...whilst World War One was an event that 'started' sometime in 1914, lasted about 4 years and 'finished' in 1918. Currently, consensus is growing that the earth is experiencing a 'global-warming' event...that 'began' somewhere in or about the 'Industrial Era' event, with geological evidence suggesting that it will end when the next cooling cycle begins. By comparison, on a smaller scale, the toss of a coin starts more or less from the moment it is flicked into the air, hopefully spends its duration spinning to the satisfaction of the observers, then ends by coming to rest on one side or the other on a hard flat surface. On an even smaller scale, a photon, arriving at the retina of an eye, starts its interaction when the 'leading edge' of the wave of its influence crossed the cellular boundary, and lasts until the entire 'pulse/quanta' has been absorbed by the biochemicals. In all such examples, determining consensus values for their 'beginnings', 'durations' and their 'ends' is highly dependent on both available technology, and intellectual perspective.

At the outset, it needs to be noted that a 'hypothetical event' is not the event that is being hypothesized...any more than a hypothetical car is a car. A 'hypothetical event' is a concept in the mind of an awareness. There may be an on-going mental event in the mind of the hypothesizer...which had a beginning and a duration and maybe even an end... but that is not the actual physical event that is being contemplated as possible.


When an event is described as having happened 'by chance' or 'randomly', we imply that such an event was unforeseen or unpredictable...either in terms of outcome, or in terms of when it might happen, or both. We are admitting that we are not aware of any guiding, causal, or influencing phenomena that would offer us the opportunity either to predict what the outcome would be, or to predict when it would occur.
On the other hand, for an event to be predictable, any causal processes must have been detected and untangled and modeled in such a manner that a prophecy process is acceptably reliable.
To our ancestors, for example, comets originally appeared in the sky by 'chance'... moved across the sky thru the stars over a period of days or weeks...and then were gone. We could predict more or less what they would look like, but had no idea at all when they might appear. It was only relatively recently in human evolution, as the model of the solar system became more closely related to observation, that it was realized that many comets follow an elliptical orbit around the sun, and that with sophisticated mathematics, it was possible to predict the return of a comet, well before its apparition.
In general, periodic events like the re-appearance of comets, heartbeats, winter and income-tax demands, are not considered to have occurred by 'chance'...because we think we can predict them. The sun did not rise in the morning by 'chance'. The last high tide did not occur fortuitously. The first flock of godwits to arrive at the mudflats in September was not a random dispersal event... they had been expected for several days.
Referring to an event as having happened by 'chance' therefore, is usually done in circumstances where it is either very unusual, has never happened before, or because we are unaware of any processes or mechanisms whereby such an event might have been caused and thereby have been able to predict it.

One of the assumptions of scientific research however, is that the methodology it uses has the power to winkle out the causal influences of any event (or sequence of events). Although this assumption has been verified over and over again, it is never-the-less the case that there are many events which we may never be able to predict with much credibility. Earthquakes, volcanic eruptions, and violent storm events are some of the most intransigent. Thus we are obliged to treat them as 'random' to a considerable extent, until such time as technology and research allow us to adjust our perspective. There may not be many phenomena at all in nature that are intrinsically 'random', and all our probabilistic mathematical modeling is a stop-gap fill-in, until technology, inclination and perseverance manage to uncover a complexity of causal links. Nature presents as chaos in many circumstances, but this is a chaos of infinite complexity, not of random and causeless happenings. We can sample nature 'randomly', but nature itself may not actually be intrinsically random in many circumstances.

On the other hand, coins and dies and cards and numbered balls and roulette wheels are all contrived artifacts of human game playing, and are specifically designed so that only one outcome event of several possibilities can happen at any one time. They are specifically designed to try and ensure that none of the potential outcomes are predictable, and thus encourage punters to risk their assets and gamble on the outcome.


For the purposes of gambling and games where an element chance is included, humans specifically design systems which have the potential for multiple unpredictable output happenings. They are designed so as to be able to predict more or less when the event will occur but not what the outcome will be.

If a system has the potential to output any one of two or more different outcomes... only one of which can occur at any one time...then the described set of hypothetical possible happenings are classified as being 'MUTUALLY EXCLUSIVE'. The described set of all possible hypothetical outcome events is called the 'Sample Space' of those outcomes. Human gaming systems are carefully designed to ensure that the 'mutually exclusive' circumstance is unambiguous and all the possible outcomes are 'equally likely'... or of at least known likelihood.

Thus for example, a cubic die with numbered faces is constructed as accurately as possible in geometric form by using a uniform and homogeneous rigid material. When this object is rolled enthusiastically across a hard horizontal surface in a uniform gravitational field, it is assumed that it is not influenced by any bias or irregularity, and that it will rotate many times about its three axes, before finally coming to rest on one of its 6 faces... in a relatively short period of time. If there is a consensus rule, that the outcome of such an event with this system is the number on the top surface of the die, then only one outcome of the six possible can occur for any one particular roll.
If we use the letter 'S' to stand for the set of all possible mutually exclusive outcomes when a numbered cubic die is rolled'... then
S(die) = { 1, 2, 3, 4, 5, 6 }
is the sample space for hypothetical die throw happenings.

As another simple example, a coin is usually constructed as a very flat cylindrical metal disc, with unique identifying markings on the two flat circular surfaces. When such a physical device is spun in a gravitational field and allowed to drop onto an (infinite) hard horizontal flat surface, the possibility of it ending up in a final configuration other than lying flat on either its 'head' side or its 'tail' side is...for all practical purposes... non-existent. It is true that hearsay and rumour would suggest that coins have miraculously actually ended up on their edge, but such a possibility is so improbable that no-one, absolutely no-one would bet on it. The top surface will be either a 'Head' or a 'Tail' and it cannot be both...(and of course, it cannot be something else.)
S(coin) = { Head, Tail }
is the sample space for hypothetical coin-toss happenings.

More complex sample spaces can be formed by linking two or more simple systems of the type outlined above. If the output of the two simple systems mentioned above were to be induced at the same time, then each of the two hypothetical outputs of the 'coin-toss' system could quite possibly occur with each of the 6 outputs of the 'die-throw' system, so that in this case there would be 2x6 = 12 possible hypothetical happenings. Such a situation can be illustrated with a simple table of possibilities.

The paired values in the above table are also 'mutually exclusive' and the set of 12 possibilities forms the 'sample space' of a possible dual 'coin-die' throw event.
S(coin-die) = { H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6 }
This sample space specifies the exhaustive set of mutually exclusive hypothetical possibilities, about what could happen if a physical 'coin-die' system was actually used. These possibilities are NOT actual events. They are hypothetical possibilities. There is only ONE actual event...and that is what happens in time and space.

[The approach that defines an event as a subset of a sample space seems particularly unwise and frought with philosophical confusion.]

In the real world, it appears to the observer that some happenings have nothing whatsoever to do with one another. The 'Triassic period' event for example would appear to have nothing to do with the 'World War One' event. A flower germination event would not be expected to have any effect on a die-throwing event. On the other hand, the latest volcanic eruption event is fairly obviously closely related to the habitation destruction events that are happening in and around the growing crater. We assume that the environment destruction events are dependent upon the volcanic eruption event. When there is no (known) causal relationship between event A and event B, we are inclined to classify them a being 'independent' events.
Multiple events can either take place at the same time or sequentially over a period of time. Two coin-toss events can be achieved either by tossing two coins at the same time, or the one coin twice in sequence. The same situation applies to a die-roll system, or any other mutually exclusive hypothetical output systems like roulette wheels or packs of cards. In the artificial contrived world of coins and dies and roulette and cards, we assume...often quite naively...that there is no possible causal relationship between the one usage of a system and another... so we presuppose that 'honesty' prevails, that there is no hidden bias, and that the output happenings of these systems are 'independent' events.


The matters outlined above have been thought about... one way or another...for hundreds of years, and as in any other arena of interest, various perspectives have been proposed and developed. The 'frequentist' focus of these considerations is purely pragmatic...which is strongly aligned with the needs of experimental science...and uses numerical and mathematical methods to find and describe patterns in nature.

To this end, the idea of 'probability' was some sort of numerical measure of the likelihood that a conceptually constrained outcome would occur.

When a human contrived physical gambling system is have several mutually exclusive hypothetical of the requirements is that when such a system is actually trialled, and ONE of the hypothetical outcomes actually occurs, there is the confident assumption that any of the other pre-trial hypothetical outcomes might have occurred. The ultimate test of any design of course, is that it must work in practice. Nothing works until it works. No design works 'hypothetically'...there is always too much to go wrong and not to have been allowed for. Repeated trials are the only practical means of checking. Repeated trials accumulate actual events, and the pattern of these actual happenings can be analyzed in order to assess whether or not all the hypothetical outcomes actually do occur enough times to justify the assumption that any of them might have happened.

When a physical system...that has been designed to have 'n' equally likely mutually-exclusive physical prepared for an experiment, the Sample Space can be represented using set notation
S(event) = { c1, c2, c3, }
If a conceptual constraint is imposed on the outcome event, and the number of elements in the sample space that satisfy that constraint is 'm', then the 'Probability' of that constraint being satisfied when the actual event occurs is given by

P(constraint) = m

The simple systems considered so far can be used to illustrate these ideas.

▶ For a 1-trial 'coin-toss' system there are n = 2 elements in the sample space
S(coin-toss) = { H, T }
If the conceptual constraint on the outcome event is "that it should be a 'Head'", then there is only 1 element in the sample space that satisfies that m = 1.
P(Head) = 1 = 0.5
The 'frequentist' interpretation of this statement is that if and when an unlimited number of trials was ever undertaken, our expectation is that the proportion of 'Heads', would be about 0.5.

If the conceptual constraint on the outcome event is that "it should be either a 'Head' or a 'Tail'", then there two elements in the sample space that satisfies that m = 2.
P(H or T) = 2 = 1
Since 'Head' and 'Tail' are the only elements in the sample space, it is impossible to get anything else, so we are obliged ot interpret P(constraint) =1 as absolute certainty.

On the other hand, if the conceptual constraint was that the outcome should be "neither a 'Head' nor a 'Tail'", then there are no elements at all that satisfy that constraint so m = 0.
P(notH or notT) = 0 = 0
thus we interpret P(constraint) = 0 as being impossible.

▶ For a 3-trial 'coin-toss' system there are n = 8 elments in the sample space

If the conceptual constraint on the outcome event is that "the first two trials must be 'Tails'", then there are 2 elements in the sample space that satisfies that constraint...TTH and m = 2.
P(TT?) = 2 = 0.25
The 'frequentist' interpretation of this statement is that if and when an unlimited number of trials was ever undertaken, our expectation is that the proportion of 'Tail-Tail-anything' results would be about 0.25.

▶ For a compound 'coin-die' system outlined above, there are n = 12 elements in the sample space.
If the conceptual constraint on the outcome event is that "there must be a 'Head' and a '6'", then there is only 1 element(green) in the sample space that satisfies that m = 1.
P(H6) = 1 = 0.0833...
If an experiment with no limit to the number of trials was undertaken, we would expect that about 83 of the 'H6' events would occur every 1000 trials.

▶ For a compound '2-die' system there are n = 36 elements in the sample space.
If a conceptual constraint on the outcome event is that "throw Y must be a '5' and throw X must be a '3'", then there is only 1 element(green) in the sample space that satisfies that m = 1.
P(Y=5 and X=3) = 1 = 0.02777...
If a conceptual constraint on the outcome event is that "X + Y < 7", then there are 15 elements(yellow) in the sample space that satisfies that m = 15.
P(X + Y < 7) = 15 = 0.4166...

There are a variety of general symbolic relationships that can be derived for when the occasion might demand. Notice that in the above example that
P(Y=5) = 1 and P(X=3) = 1
6 6
So that in this case

P(Y=5 and X=3)     =     P(Y=5).P(X=3)

That is... the probability of both events occurring is equal to the product of the probabilities of each event happening separately.

This is a general result for independent events and can be illustrated as follows. Every one of the hypothetical outcomes of a system A can be associated with every one of the outcomes from a system B.
The table below shows the entire two-event sample space.
Taken separately, since there are n hypothetical elements in the sample space of system A, the probability of any one of them actually occurring is (1/n), and similarly, the probability of any one of the possibilities of system B is (1/m)...but considered as a single two-event outcome however, any one of the possible two-event pairs is a member of a sample space with (nxm) elements, so the probability of that occurring is (1/mn).
P(ai) = 1 and P(bj) = 1
n m
P(aibj) = 1
n x m

so that   P(aibj)     =     P(ai).P(bj)

In dealing with such problems it is important that all the possibilities are identified for both the entire sample space and the outcome constraints, and not to jump to hasty conclusions. Consider the following example:

▶ Suppose we have a designed system which consists of a box in which there are 4 red balls and 3 black balls all covered over with sawdust, and that we have provided for an efficient mixing component that will ensure a thorough chaotic redistribution of the contents before/after every trial.

If we carried out an unlimited number of 'lucky-dip' trials of selecting 2 balls after the other without replacement... what would our prediction be as to the ratio of successes?

At first glance, there is a temptation to simply look at the situation, note that there are 7 balls to choose from, that we are going to choose 2, and so that the 'probability' would be have to be 2/7= 0.286 would it not?

But this assumption would be quite inaccurate, because the constraints of the situation have not been analyzed with sufficient care, and we will find that experimental trials disagree. Actually physically repeating the experiment many times is very time-consuming and does not result in any reliable trends until very large numbers of trials have been achieved. Using computer simulation therefore, is a way of acquiring some confidence about what would actually happen if indeed we could find the time and inclination to conduct the experimental check. Computer simulations of 1000 trials resulted in a graphs of success ratio that often looks something like the one below.

Ten of these simulations were run...amounting to 10000 trials...
giving final ratios of 0.142, 0.133, 0.126, 0.15, 0.149, 0.144, 0.14, 0.144, 0.142, 0.138... the average value of which was 0.1408.
It looks very likely that the assumption P = 2/7 = 0.286 is not correct.

What we need here is at least one and preferably two or more different analytical approaches that 'predict' the actual outcome of the real-world experimental situation.

Firstly, let us try the 'two-event' analysis outlined above. When 2 balls are drawn from the box system one after the other, they are in fact two different events, and almost without noticing, the system for the first draw is different to the system for the second draw. The first system has 7 balls in the box, but the second system only has 6 balls in the box.

For'success' both balls must be black so the first draw from system A must be black with a probability of 3/7...since the sample space has n = 7 elements and the number of elements satisfying the constraint of being black m = 3. For continued success the second draw from system B must also be black, but now, one of the black balls is missing. The probability of this second event being a black ball is now 2/6...since the sample space for system B has n = 6 elements and the number of elements satisfying the constraint of being black is m = 2. There does not appear to be any obvious reason why the event of drawing a ball from system A should influence the result of drawing a ball from system B, so we should be able to assume that the two events are independent.
Thus P(Black then Black) = P(Black in system A)xP(Black in system B)
P(Black then Black) = (3/7) x (2/6) = (1/7) = 0.14285
This certainly agrees much better with our simulation results.

Secondly, as a different line of approach, let us try and analyze what the sample space would be for a hypothetical two ball draw if the coloured balls were numbered as well.

When two balls are drawn, we were only interested in their colour, so it doesn't matter what the order was. All we need to discover is how many ways two balls can be selected from seven, when order is not important, and one ball cannot be selected twice. We cannot select red ball 1 then red ball 1 again, because replacement is not allowed. Selecting red ball 1 then red ball 2 is essentially the same outcome as selecting red ball 2 then red ball 1, so there is no need to count these duplications.
A systematic table of possibilities might look like this.

invalid 1 - 2 1 - 3 1 - 4 1 - 5 1 - 6 1 - 7
[2-1] invalid 2 - 3 2 - 4 2 - 5 2 - 6 2 - 7
exclude exclude invalid 3 - 4 3 - 5 3 - 6 3 - 7
exclude exclude exclude invalid 4 - 5 4 - 6 4 - 7
exclude exclude exclude exclude invalid 5 - 6 5 - 7
exclude exclude exclude exclude exclude invalid 6 - 7
exclude exclude exclude exclude exclude exclude invalid

These n = 21 possibilities therefore are an exhaustive list of all unordered two-ball choices without replacement, and so is the sample space of the system A.
Further examination of the table immediately reveals that the only possible two-black choices are ( 5 - 6), ( 5 - 7 ) and ( 6 - 7 )...i.e. the number of possible elements in the sample space that satisfy the 'being both black' that m = 3.
Thus     P(Black then Black) = m/n = 3/21 = 1/7 = 0.14285.
which agrees exactly with the 'two-event' analysis described first.

There are at least two other analytical methods...using ideas of 'permutations and combinations' and ideas of 'conditional probability'... that result in exactly the same probability prediction of 1/7 for the ratio of successful two-black-ball draws. The main issue that emerges from this exercise however, is not the veracity or otherwise of the analytic prediction methods, but the magnitude of the number of 'trials' that appear to be needed before the predictions can be confirmed or not. Computer simulations are not actual trials, no matter how much we persuade ourselves that they are valid, and the 10,000 simulated trials of the above system would only rarely...if ever...actually be undertaken. The problem is anything but trivial, which we can illustrate by conducting real physical trials on the very simplest of systems...the coin-toss.


Performing multiple trials of mutually exclusive output systems raises problems of proliferation and interpretation as the complexity of the systems increase. When a physical system with N mutually exclusive hypothetical outputs is trialled once, there is only one output event happening.
One toss of a coin system results in one event...(of the N =2 possibilities)
One throw of a die system results in one event...(of the N = 6 possibilities)
When multiple events from such systems are required, the most practical approach is usually to repeatedly use one system many times...rather that try and create huge numbers of identical physical systems. Multiple trials provide repeated opportunities for each of the potential output options to occur, and experience at trialling such systems gives rise to certain expectations. At first sight...with persistence and no limit to the number of trials that can be seems to be that all of the different possible outputs will eventually occur and continue to occur, but very real practical limitations quickly become evident.

Consider the simplest of all mutually exclusive outcome systems...the 'coin-toss'...
not some 'computer-simulation' of the system, but an actual (metal) coin stamped differently on each side, and spun-tossed 'randomly' by a person so as to land on a flat surface.

▶ For 1-coin toss trials, the sample space of possibilities is simply
S(1) = {H,T}
There are only the two possibilities, and one would expect from experience, that both the 'Head' and the 'Tail' would occur sooner or later when successive trials are carried out. Conducting such a trial right now... on the spot...the sequence of trial event outcomes was...
T, T, H, T, H, T, T, H (stop)
wherein both possibilities happened at least once as expected.

▶ For 2-coin trials there are 22 = 4 possibilities.
which can be accounted for systematically on a table thus...

1st Trial 2nd Trial

There were only two possibilities on the first trial, but whichever it was, there were 2 possibilities on the second trial. The table can now be rewritten to show the 4 different possible outcome pairs.

1st Trial 2nd Trial

We can now write the sample space as a list of set members and be confident that we have not missed one.
S(2) = {HH,HT,TH,TT}

Conducting another 'on the spot' trial for this sample space, it took 9 trials until all possibilities were obtained at least once ...
TH, TT, TH, TT, HT, TT, TT, TT, HH (stop)

▶ For 3-coin trials there are 23 = 8 possibilities

1st Trial 2nd Trial 3rd Trial

So now we can confidently list the exhaustive sample space.


The first few on the spot trials were...
I had to do at least 8 trials of course, but after 13 trials, boredom took over and I stopped, and there were still 2 of the possibilities that had not occurred. You could do your own trials and not give up so easily, and we could both be confident that all of the possibilities would happen within an hour or so no matter how exceptional a set of trials you obtained.

The point of these little exercises is TIME. It takes time to conduct these trials and as the number of events in each trial increases, the amount of time required to check what actual outomes compared to the hypothetical sample space...becomes far more than one might have ever imagined.

▶ For 4-coin tosses there are 24 = 16 possibilities

From the little experimenting we have done, it would seem quite likely, that conducting a trial series of 4-coin tosses might take quite a long time before all the possibilities occurred. Sharing the load, and getting several coin-toss systems running in parallel would speed things up of course, but the problems of reality are looming.

▶ Suppose now, we cut to the chase and precipitate the magnitude of the impending reality. Suppose we conducted a 50-coin toss trial.
Here is my first trial...

Now there are actually 250 = 1,125,899,906,842,624 possible happenings in the sample space for this sort of multiple event, of which the trial above is only ONE of them!
At this point, to get things in perspective, we might note that this number is about the same size as the number of seconds in 50 million years...i.e. 1,577,880,000,000,000.
So if one of these 50-coin trials could be done every 100secs, it would take about 5 billion years...(the estimated age of the universe), to even do the same number of trials as there are possibilities in the sample space. Clearly these considerations are impractical and absurd and require some adjustment of perspective.


If I take the 165 Head/Tail events that have been produced here so far in these few multiple trials and consider them as a sequence in time, one after the other, we can keep a cumulative score of each of the Head and Tail events, and compute the ratio of each with respect to the total number of trials. We put the results into a spreadsheet, obtain a graph, and output as a .jpg file.

Because this is a binary system, there is a symmetry about the two time-line graphs. If it wasn't a Head, then it would have been a Tail, and vice versa, so a little reflection will reveal that the cumulative sum of the Heads and Tails will always be equal to the total number of trials, and the two ratios will always add up to one.
What immediately gets our attention is the fact that the two ratios flirt with the value '0.5' = 1/2. The Heads is 0.461 and the Tails is 0.539.
When we think about this our attempts to find numerical patterns in unpredictable chaos...we speculate about what might be the consequences of extrapolating the 'experiment' that we have conducted so far, and are quickly confronted with a dilemma. On the one hand, we would like to assume that future coin-toss events will be 'equally likely', and therefore the ratios of Head and Tails will both get closer and closer to '0.5'...but on the other hand, we are obliged to admit that no sequence of events is 'forbidden' or 'impossible', and so a run of 1 million heads in a row 'could' happen. What to do?
One strategy would be to assume the 'equally likely' stance at all costs as an intellectual principle. If an unexpected unlikely series of events actually occur, we grit out teeth, firm up our resolve, and continue the experiment in the face of all adversity, in the unshakable conviction that 'eventually' our assumption will be justified. This is the gamblers 'casino' approach, where in spite of unbelievable 'bad luck', the certainty endures, that eventually...eventually...the luck will change, and everything will get back to even keel. The reality is however, that this approach is often impractical or impossible because of the limitations of time.
The pragmatic strategy is to adopt the 'equally likely' stance, presume the 'highly unlikely' will not happen, and that eventually, the Head and Tail ratios will get as close to '0.5' as we are prepared to attempt. This is not dissimilar to making ones home on the side of an 'extinct' volcano. On a human timescale, an eruption is very improbable, but if it does, you have to deal with the consequences.