That said let's consider the system (one roll of a dice) mechanistically. We know from basic physics that given all the information (force, direction, mass, etc.) of the system (one roll of a dice) we can predict the outcome with perfect accuracy. In other words the system (one roll of a dice) is, well, deterministic (certainty assured). — TheMadFool

How do you figure this? Practically speaking, physical science is always subject to some degree of inaccuracy.

How do you figure this? Practically speaking, physical science is always subject to some degree of inaccuracy. — Pantagruel

Physics/mechanics???!!! We've put men on the moon. Surely a humble dice is within its reach.

Physics/mechanics???!!! We've put men on the moon. Surely a humble dice is within its reach. — TheMadFool

To quote Regis, Is that your final answer?

Physics/mechanics???!!! We've put men on the moon. Surely a humble dice is within its reach. — TheMadFool

It doesn’t take 100% accuracy to put men on the moon. Also modeling gravitation in space is much easier than modeling all frictions on a dice thrown in the air and bouncing on a surface: the dice will bounce differently depending on the hardness of the surface at the precise point where it bounces, and a tiny change in the angle at which the dice bounces will totally change how it bounces and its subsequent motion, so it’s a chaotic system, a tiny difference in initial conditions will change the final state of the dice and in most cases we can’t measure all relevant variables with sufficient accuracy. Also, the guys going to the moon could control their trajectory to some extent during the flight, whereas we don’t have little guys controlling and stabilizing the dice while it flies and bounces :wink:

What causes an unequivocally deterministic system to exhibit probabilistic behavior? — TheMadFool

Exhibit is the keyword here. What is exhibited is in the eye of the beholder - it is not just an objective property of a system. A system may hypothetically be perfectly deterministic (although how could you know that with certainty?), but if you don't know enough about its behavior, then you can, at best, predict it probabilistically. Typically, when a die is thrown, there is no practical way to predict its exact trajectory (even if there was a fact of the matter about what that trajectory would be), nor even which side it is more likely to land on. Lacking such information, the prudent bet is to distribute probability equally between each of the six sides. (If you systematically fail to do this, then a competing player could exploit your bias to gain an advantage.)

1. Is probability an illusion? — TheMadFool

Calling it an "illusion" implies that you know better. But you don't - hence probability. Probability is a function of our uncertainty.

To quote Regis, Is that your final answer? — Pantagruel

Is there any other answer? Look, I'm willing to accept that there is an issue of scale when it comes to laws of nature. For instance the quantum world is claimed to be fundamentally different from the world of suns, planets and galaxies and coincidentally what sets them apart is that the quantum is probabilistic; not so suns, planets and galaxies.

However, a dice and a planet or you or me seem to be within the range of the mechanistic physical laws of Newton. Measures of force, mass, angles, and other relevant data are all we need to precisely predict what the outcome of dice-roll will be. Kindly read my reply to Leo below:

It doesn’t take 100% accuracy to put men on the moon. Also modeling gravitation in space is much easier than modeling all frictions on a dice thrown in the air and bouncing on a surface: the dice will bounce differently depending on the hardness of the surface at the precise point where it bounces, and a tiny change in the angle at which the dice bounces will totally change how it bounces and its subsequent motion, so it’s a chaotic system, a tiny difference in initial conditions will change the final state of the dice and in most cases we can’t measure all relevant variables with sufficient accuracy. Also, the guys going to the moon could control their trajectory to some extent during the flight, whereas we don’t have little guys controlling and stabilizing the dice while it flies and bounces :wink: — leo

Ok. Let's suppose that a normal-sized dice is a "chaotic system" and is actually probabilistic. Just blow-up the dice - increase its size to that of a room or house even. Such a dice, despite its size, would continue to behave in a probabilistic manner despite our ability to predict the outcomes accurately. After all you do accept that rocket trajectories can be predicted and therefore controlled.

Exhibit is the keyword here — SophistiCat

Yes. My issue is that a deterministic system performs like a probabilistic one. It's not an issue of incomplete data for computation as an enlarged dice, something we can have adequate data on, will continue to be probabilistic. We can even calculate the probabilities of every possible outcome which will match the experimental results. Yet, such a system (enlarged dice rolling) is in fact deterministic.

To All

Why does a system whose outcomes we can actually predict behave as if we can't do that? That's what bothers me. Thanks.

In my understanding you are somehow missing certain points.

For quantum events as far as I am informed they are understood to be probabilistic in nature. However the using of bigger scales doesn't negate this it only creates a state where events thought of as inpossible in the mechanistic view have a very low probability.

However lets ignore that considering a chaotic system the scale of the system shouldn't have an influence on it being chaotic or not. Chaotic means as others have stated before a sensitivity to input/relevant factors.
This means in this example that if you role a dice with a specific force that leads to a certain result the increasing of the force by 0,00...1 can lead to a completley different result. Same for angle and all the other factors. I don't see how this should change based on scale.

However lets ignore this case aswell and assume a completley deterministic system.
If you have a set of factors (force, angle, ect) and you throw the dice it gives a specific result. If you reproduce the same initial conditions you create exactly the same output. This is technically still under the scope of probability by it being modeld as 100 % chance to get the same result. However if you just role the dice again without fixing the intial conditions it should be logical to conclude that different intial conditions can produce a different outcome.

Therefore roling a dice with not fixed initial conditions is not clearly determined and thus probabilistic. Theres a chance that you use initial conditions I which lead to specific outcome O.

So somehow I don't get what your asking.
If you imagine a scenario of throwing a rock that is soley dependant on force and not chaotic at all (more force=flies further) you seem to be asking in my understanding if I throw a rock multiple times (with different force) why doesn't it always lead to the same result. (Excluding chaos). However as soon as you include chaos (different hights different density of air, relationship air particles with object) the clear predictability is gone.
So you seem to be investigating a chaotic system however meanwhile refusing to view it as chaotic and postulating it to be clearly predictable.

Did I missread you somewhere?

Probability is the mathematical study of chance which basically considers events that are uncertain. We can't make definite claims in probability. — TheMadFool

Yes we can. The ‘definite’ claim is probabilistic though. Mathematics operates within abstract hypotheticals it is not based experimentation.

If we set up a mathematic problem where the outcome is A or B we can never ever get an answer other than A or B - the probability of getting A or B is 100%.

If you’re simply stating that mathematical models do not map 100% onto reality then why are you bothering to state this? It’s obvious. I guess some mathematical model would map onto the entire universe but I don’t see how we’d have anyway of knowing this even if we happen to stumble across it by pure fluke.

Yes we can. The ‘definite’ claim is probabilistic though. — I like sushi

What is the difference between probability and certainty in your view?

I didn't give it much thought and it probably shows. Anyway...

Imagine two 6-sided dice A and B

Imagine A is rolled in our world by a person like you or me. The outcome of the dice is predictable given initial conditions. You can't disagree on that.

Now imagine dice B exists in a non-deterministic world where probability is real.

If you test dices A and B you'll observe that both yield the same results as if both resided in a non-deterministic world.

Can you explain why?

In my view? You said ‘mathematics’ so I don’t have an opinion on the matter. Certainty, in mathematics, is - for example - 1+1=2. I don’t have an opinion about this.

Mathematical probability isn’t based on observation/experimentation. It is used to interpret experimentation and observation thought aided my measurements.

Don’t conflate the abstract with the concrete when talking about mathematical models and reality.

In my view? You said ‘mathematics’ so I don’t have an opinion on the matter. Certainty, in mathematics, is - for example - 1+1=2. I don’t have an opinion about this.

Mathematical probability isn’t based on observation/experimentation. It is used to interpret experimentation and observation thought aided my measurements.

Don’t conflate the abstract with the concrete when talking about mathematical models and reality. — I like sushi

Well you said

Yes we can. The ‘definite’ claim is probabilistic though — I like sushi

The word "definite" is usually associated with certainty which is the antithesis of probability. I thought you had an interesting take on the subject.

Actually I think we can calculate "exact" probabilities e.g. in the chance of getting a heads on a single coin-flip is "exactly" 50%. No more, no less.

However, I'd like you to take a step back into the nature, as generally understood, of probability which is basically an uncertainty about a given event. For instance we have a "definite" number, say 70%, when meteorologists predict the weather. Yes, I'm sure about 70% - it's a definite quantity - but are you certain that it'll rain or not?

Actually I think we can calculate "exact" probabilities e.g. in the chance of getting a heads on a single coin-flip is "exactly" 50%. No more, no less. — TheMadFool

Nope. As good as 50% but not exact. For starters force of coin flip, wind factor and the weight distribution of the coin are all physical factors in the real world - not to mention the rare occasions where a coin lands on its edge.

Granted, in day-to-day speech we refer to a coin toss as being 50-50.

Yes, I'm sure about 70% - it's a definite quantity - but are you certain that it'll rain or not? — TheMadFool

If you cannot answer that question yourself I don’t think I know what you’re trying to talk about? ‘Definite quantity’? This is just word play isn’t it. Giving an ‘exact’ number doesn’t mean anything out of the context it is given in.

Honestly, I’ve no idea what you’re talking about. Sorry :(

I think the best answer I can give you is that materials science if by it's nature inexact, due to the innate complexities of natural objects (systems) which limit our ability to perform exact measurements which precisely determine their properties. Sure, we could attempt to model these objects as non-linear dynamical systems, but in the end all that would be doing is replacing probability with systems theory.

Yes. — TheMadFool

No, you still don't understand. Suppose the die already landed and came to rest, but you can't see it - it rolled under a couch. The outcome is not just deterministic - it's already determined. Does that fact help you with guessing the answer? It doesn't, and you still don't have anything better than probability. So would you say that a die lying on the floor under your couch "exhibits probabilistic properties?" That would be stupid (but what else is new?)

Probability is about what you don't know. Whether what you don't know is inherently indeterministic or whether it is a matter of fact doesn't make a difference to you. You still don't know what you don't know. So what do you do? You could say "I don't know," but if you have to make decisions based on incomplete knowledge, you use probability.

Ok. Let's suppose that a normal-sized dice is a "chaotic system" and is actually probabilistic. Just blow-up the dice - increase its size to that of a room or house even. Such a dice, despite its size, would continue to behave in a probabilistic manner despite our ability to predict the outcomes accurately. After all you do accept that rocket trajectories can be predicted and therefore controlled. — TheMadFool

If we somehow launch the house-sized dice fast enough it isn’t clear that we could predict individual outcomes accurately, because again an extremely tiny change in how the dice is thrown or in the wind or in exactly where and at what angle it bounces would change the outcome. While if we launch it so slowly that the dice doesn’t rotate, we would predict the outcome but then it wouldn’t be probabilistic anymore.

Rocket trajectories aren’t predicted with 100% accuracy, in the atmosphere their trajectories are constantly corrected and stabilized in order to stay on a given course, and in space they are also corrected every now and then, because they encounter accelerations (due to various phenomena: wind, dust, radiation pressure, ...) that aren’t predicted beforehand with 100% accuracy.

However it is possible to have systems where overall the outcomes are overall probabilistic even though we can predict each outcome individually. Consider a simple system where you throw the dice so slowly and at such low height that it doesn’t rotate while in the air and it doesn’t rotate after bouncing. Then you can always predict what the outcome will be, but in order for the outcomes to be probabilistic you have to change the initial conditions probabilistically (namely the initial orientation of the dice). So if 1/6 of the time you start with 1 up, 1/6 of the time 2 up ... and so on, you know that the dice will land 1/6 of the time with 1 up, 1/6 of the time with 2 up and so on.

It is also possible to have systems where you don’t control the initial conditions (contrary to the above example), where you can predict each individual outcome and where the outcomes are still probabilistic overall. For instance let’s say you pick 1000 people randomly (from various places without looking at them) and each of them writes their age on a piece of paper, so you have 1000 pieces of paper with some number on them. Now say you pick 10 pieces randomly, you add the 10 numbers together and you divide by 10, that gives you the average of these 10 numbers, let’s call it N1. You put the 10 pieces back with all the other and again you pick 10 pieces randomly, you carry out the same process and you get another number, N2, and so on and so forth, you do that many times. Each time you pick 10 pieces, the outcome number can be predicted (you just have to take the sum and divide by 10), however it can be shown mathematically that the numbers N1, N2, N3, ... follow a probabilistic distribution, a so-called normal (or Gaussian) distribution, that’s to say that most N values will be gathered in a tight range, and the further away from that range the less and less values there are, and you can compute how probable it is for a random N value to fall into such and such range. So before you pick 10 pieces of paper again, you know how likely it is that their average will fall into such and such range.


Going back to the example of the dice, is it possible to not control the initial conditions, to be able to predict each individual outcome and yet the overall outcomes are probabilistic? It is possible in some specific cases:

What has an influence on the dice is the initial conditions (the orientation of the dice when it is thrown and the velocity and angle at which it is thrown), and the forces acting on the dice during its motion: air friction that depends on the initial conditions, on air density, on wind; bouncing force provided by the surface on which the dice bounces that depends on the initial conditions, on the local hardness/elasticity of the surface, on the shape of the surface as well as that of the dice; and surface friction when the dice is rolling on the surface, which also depends on initial conditions and on the surface itself as well as that of the dice

If the air density, air wind, and the properties of the surfaces didn’t change, then that means that if you launched the dice several times with the same initial conditions, you would always get the same outcome. So if these properties didn’t change, in order for the outcomes to be overall probabilistic, the initial conditions would have to be probabilistic (so you would have to always throw the dice differently). Since the forces acting on the motion of the dice depend on the initial conditions in a complex enough way, a tiny change in initial conditions will be enough to usually change the outcome. And since the dice can only land in 6 possible ways, we can expect to get each side about 1/6 of the time.

Otherwise, if you always launched the dice with the same initial conditions, in order for the outcomes to be probabilistic the air density/wind and/or the properties of the surfaces would have to change between each throw, but then it would be impractical to accurately measure these evolving conditions and so it would be impractical to predict individual outcomes.

So in practice, if you conduct the experiment in a closed system where the air density is constant, where there is no wind besides that generated by the dice, and where the surfaces are hard and regular enough (both the surface of the dice and the surface on which the dice bounces), then it is possible to predict individual outcomes and to have overall probabilistic outcomes only if you always change the way you throw the dice (otherwise if you throw it in the exact same way you are guaranteed to get the same outcome and then the overall outcomes won’t be probabilistic).


And now to finally answer your questions, notice that in both this last example and in the example with the numbers on pieces of paper, any individual outcome is deterministic, but the way these outcomes are distributed is probabilistic. In the last example of the dice, in all the ways that the dice can be launched, in about 1/6 of the case the dice lands on a given face, because the dice can only land in 6 different ways, and the forces acting on the dice are complex enough that they don’t lead to one side being more likely than the other, whereas if the forces were much simpler (like in the example where the dice never rotates during its motion) then one side would be preferred. In the example of the pieces of paper, it can be shown mathematically that the outcomes are distributed in a probabilistic way, following a Gaussian probability distribution: that’s called the “central limit theorem”.

In deterministic systems probability is not fundamental in the sense that if we can’t predict an individual outcome it’s only because of incomplete knowledge. As to your second question, the deterministic system does not exhibit fundamental probabilistic behavior, a more correct way to phrase it is that it exhibits statistics, due to the configuration of the system itself. In the dice example it is possible to configure the system in a way that some outcome is preferred, for instance always starting with the same initial conditions. Or we could start with different initial conditions but make one side of the dice more sticky than the others so that the dice will land more often on the sticky side.

In deterministic systems probabilities are not fundamental, rather we compute statistics that depend on the configuration of the system, which can be interpreted as a characteristic or a property of the system. The probabilities in deterministic systems refer to incomplete knowledge, so in the example of the dice where the outcome can be predicted, we would say the dice has a given probability of landing on a given side when we have incomplete knowledge of the initial conditions or of how the system reacts to these initial conditions.

Say there are N different possible configurations of initial conditions, and we know the system always reacts the same to a given initial condition (because in that particular system when we precisely measured the initial condition and we kept it the same we always got the same outcome), and we know that in about N/6 configurations the dice lands on a given side (because we have thrown the dice a great number of times in that system with different initial conditions and that’s what we have noticed), but this time when we throw the dice we don’t measure the initial condition (the initial velocity/angle/orientation of the dice), then we say that the dice has about 1/6 probability of landing on a given side, but that’s only because we don’t know in which deterministic configuration we are once the dice is thrown, because we haven’t carried out the necessary measurements that would provide us with that knowledge.

And depending on the system the number doesn’t have to be exactly N/6 for each side, in most deterministic systems it wouldn’t be exactly N/6, depending on the dice and on how the system reacts to it.


That turned out longer than I expected, hope that helps.

I disagree.

If we keep the exact same initial conditions A will always produce the same result unlike B that will produce differenig results due to its indeterminsim.

So I disagree that they produce the same result.

It is not probability that is an illusion; it is certainty.

There is no justification for certainty in our perception of the real world. Any claims for certainty are either a subjective convenience or an illusion.

That sums it up well enough :)

"Why does a system whose outcomes we can actually predict behave as if we can't do that? That's what bothers me. Thanks."
In an ideal world, human knowledge would contain complete understanding of how the universe works.
Lacking that, we rely on models, representations, and approximations to the physical behavior of the world. The idea that knowing the current state of the universe, by knowing the positions and motions of every element, is an impossibility, since that awareness is always historical.
The rules (approximations of laws) for tossing dice are incomplete. Predictions are based on statistics, a history of past events.

Reality isn't deterministic, aka we cannot accurately predict the future ever, because of emergence and flux (everything influencing everything else). So, you also have an undeterminable probability of the success or failure of your mechanistic prediction.

That turned out longer than I expected, hope that helps. — leo

No, you still don't understand. — SophistiCat

Honestly, I’ve no idea what you’re talking about. Sorry :( — I like sushi

I apologize for lumping you all in a group but all of you deny that there is a problem here. Maybe as Sophisticat said I've misunderstood but I hope to express my thoughts more clearly in this post.

Firstly we must agree that we can predict, using physics, the outcomes of events at the human scale. Set aside the complexity of the issue for the moment and consider that given all initial values of a system the outcome pathway is fully determined. leo said that this isn't possible with 100% accuracy which I disagree to. Take the simple example of a space probe. Using rockets in the right locations and fired for the correct durations we can and do predict that the probe lands right side up. This is a 100% prediction accuracy and if there are any inaccuracies they are due to unforseen cotingencies like wind or instrument malfunction.

I hope we can agree now that physical systems at the human scale are deterministic and that includes a fair 6-sided dice which I want to use for this thought experiment.

Imagine there are two people A and B with one fair 6-sided dice. A throws the dice but is oblivious of the initial state of the dice. B has complete knowledge of the initial state of the dice. As you see A can't predict the outcome but B can.

The theoretical probability that the dice will land on a number less than 5 is 4/6 = 2/3 i.e the dice should show a number less than 5 two-thirds of the time (66.66%)

We are now going to calculate the experimental probability. A throws the dice 100 times and checks how the dice lands.

Remember that each time A throws the dice, B can predict the outcome of the dice. So...

The result of the experiment of a 100 throws of the dice will show that the dice lands on a number less than 5 approximately 66 times. This result is in agreement with the theoretical probability calculated (4/6 = 2/3 = 66.66%). In other words the system (person A and the dice) behaves like a probabilistic system as if the system is truly non-determinsitic/probabilistic.

However, B knows, since he knows the initial states of each dice throw, that the system (person A and the dice) is deterministic/non-probabilistic and that each outcome is predictable.

1. We know that the system (person A and the dice) is deterministic because person B can predict every single outcome.

2. We know that the system (person A and the dice) is probabilistic because the experimental probability agrees with the theoretical probability which assumes the system is non-deterministic.

There is a contradiction is there not?

It is not probability that is an illusion; it is certainty. — A Seagull

We can't say that because we know the system (person A and the dice) is deterministic because person B can predict the outcomes of every single throw of the dice.

As a wise man once said: " The world is is not only queerer than we suppose it is queerer than we can suppose.

To try to impose one's pre-supposed ideas and assumptions about the way the world 'should' be is naive.

If you still have a problem with probability and tumbling dice, I suggest you re-visit your assumptions regarding the way you think the world should be.

2. We know that the system (person A and the dice) is probabilistic because the experimental probability agrees with the theoretical probability which assumes the system is non-deterministic. — TheMadFool

I wouldn't say that theoretical probability assumes the system is non-deterministic. Rather, it assumes that the system has certain regularities that enable us to calculate frequencies of possible outcomes. In the example of throwing the dice, the regularities are (deterministic) laws of physics that transform initial conditions to outcomes.

1. We know that the system (person A and the dice) is deterministic because person B can predict every single outcome.

2. We know that the system (person A and the dice) is probabilistic because the experimental probability agrees with the theoretical probability which assumes the system is non-deterministic.

There is a contradiction is there not? — TheMadFool

If there is a contradiction, it is only in how the system is being represented. In this scenario, person B has complete information about the system whereas person A has only partial information. The difference is not in the system but in the information that each person has.

Which is to say, you can represent the system (person A and the dice) from either person A's point-of-view or person B's point-of-view, which avoids contradiction.

This result is in agreement with the theoretical probability calculated (4/6 = 2/3 = 66.66%). In other words the system (person A and the dice) behaves like a probabilistic system as if the system is truly non-determinsitic/probabilistic. — TheMadFool

And if A threw a hundred sixes in a row it wouldn't be behaving like a probablilistic system?

Set aside the complexity of the issue for the moment and consider that given all initial values of a system the outcome pathway is fully determined. leo said that this isn't possible with 100% accuracy which I disagree to. — TheMadFool

I said it’s possible in very specific cases, most of the time it isn’t.

Take the simple example of a space probe. Using rockets in the right locations and fired for the correct durations we can and do predict that the probe lands right side up. — TheMadFool

It seems you missed the part where I said that the rockets are constantly controlled and stabilized during their flight, so as to remain on the desired trajectory. That control and stabilization is not predicted in advance, it dynamically adjusts to the conditions that the rocket encounters through a negative feedback mechanism.

This is a 100% prediction accuracy and if there are any inaccuracies they are due to unforseen cotingencies like wind or instrument malfunction. — TheMadFool

Without the dynamical control/stabilization the probe would never reach its destination. The unforeseen conditions that the rocket/probe encounters are precisely what makes the outcome unpredictable from the initial conditions alone. Which is why I said that the outcome is determined from the initial conditions alone only in simple cases where everything that happens is predictable, if you read my long post carefully you will find the answers to your questions.

I hope we can agree now that physical systems at the human scale are deterministic and that includes a fair 6-sided dice which I want to use for this thought experiment. — TheMadFool

Then consider the experiments where the outcome of each individual throw can actually be predicted, that is in simple situations with no wind except that caused by the dice, constant air density, constant temperature, smooth and hard surfaces. As I explained to you in real experiments the outcome usually won’t be exactly 1/6 for each side.

If the experiment is deterministic then there are a finite different number of ways to throw the dice, a finite number of initial conditions and outcomes, let’s call it N. If you measure all the outcomes, usually each side won’t appear exactly N/6 times. As I explained, if one side of the dice is slightly more sticky than the others, that side will show up quite more often. And even with the same stickiness, consider that a dice is not exactly symmetrical due to the dots or numbers printed or engraved on the surface, so in deterministic experiments the outcomes wouldn’t have exactly equal probability.

If instead you consider experiments where there is wind, or changing temperature, or irregular and soft surfaces, you can’t predict individual outcomes, and the various forces act together in such a complex way that they don’t prefer any particular side of the dice, even starting from the exact same initial conditions.

So for your thought experiment to make sense, consider that in deterministic systems where the outcome can actually be predicted in practice, starting from the same initial condition leads to the same outcome, and in such systems the dice does not land exactly as many times on each side.

1. We know that the system (person A and the dice) is deterministic because person B can predict every single outcome.

2. We know that the system (person A and the dice) is probabilistic because the experimental probability agrees with the theoretical probability which assumes the system is non-deterministic.

There is a contradiction is there not? — TheMadFool

No, because again, as I and others have explained, the theoretical probability does not assume the system is non-deterministic. In deterministic systems probability is not fundamental, if it confuses you use the term statistics instead.

You can come up with deterministic systems in which the dice will always land on the same side, or twice more often on some sides than on some the others. In some deterministic systems the dice lands about equally often on each side, in some other deterministic systems that’s not the case at all. Consider these latter systems and that will help you understand your error.

In deterministic systems the outcome is about 1/6 for each side only when the configuration of the system is such that it does not prefer any particular side. Some deterministic systems are like that. Some aren’t.

The problem is you assume in any deterministic system where each individual outcome can be predicted the dice will always land about equally often on each side. That’s wrong. Also in most situations where we throw a dice we can’t predict each individual outcome. Try to build a deterministic system in which you can predict each individual outcome, that will help you understand your error too. In your reasoning you assume that each individual outcome can be individually predicted, then consider real systems where that’s actually the case, where that’s actually done in practice, otherwise your thought experiment isn’t connected to reality.

In some deterministic systems the dice lands about equally often on each side because in such systems the symmetry of the dice doesn’t lead the system to prefer a particular side. How often an object lands on each side depends on the shape of the object, on its symmetries. You get symmetrical outcomes when you are dealing with a symmetrical object and when the system doesn’t break that symmetry. This doesn’t mean that the deterministic system is exhibiting non-deterministic behavior, there is no mystery here.

A coin is partially symmetrical, such that it lands most often heads or tails, but it can also rarely land on its edge. Yet you can construct a deterministic system in which it always or almost always lands on its edge (make a system in which the coin bounces or slides on inclined surfaces so that it ends up in a groove the same width as the edge of the coin). In this case the deterministic system prefers a particular symmetry of the object. In that system the coin wouldn’t land heads 50% of the time and tails 50% of the time. And just because you can say that in this system the coin lands on its edge about 100% of the time, using probability jargon, that doesn’t mean that the deterministic system exhibits non-deterministic behavior, just like when a coin lands heads or tails about 50% of the time that doesn’t mean that the deterministic system exhibits non-deterministic behavior, just like when a dice lands about 1/6 of the time on a given side that doesn’t mean that the deterministic system exhibits non-deterministic behavior, with the dice too the system can be configured so that some particular side/sides is/are preferred...

The world 'appears' deterministic at times at the human scale (e.g billiard balls on a pool table) but this in fact is only an artefact of approximate perception. Is that the origin of the confusion?

It doesn’t take 100% accuracy to put men on the moon. — leo

It seems that if your goal is to put a man on the moon and you put a man on the moon, your knowledge was 100% accurate. Now, if you wanted to put a man on a certain area of the moon that is only 50 meters in diameter, then that would be a more difficult stunt to pull off. That would require more specific/relevant knowledge to accomplish.

It's interesting to note that the difficulty of some task seems to coincide with it's probability of being accomplished. The more difficult the task, the lower the probability. How difficult a task is is dependent upon how experienced we are with that task - how many times we've done it and worked out all of the kinks in our understanding of the process that it takes to accomplish the goal. It seems to me that these indicate some kind of subjective skewed view of the world where we are imposing our probabilities and level of difficulty out onto the world that isn't probabilistic or difficult/easy. It just is a certain way, which includes the amount of knowledge we have about it.

Our knowledge isn't accurate because we aren't omniscient. We don't have direct access to the entire universe at every moment of our life. What gives us the power to get close enough, if not all the way, is mathematics. Mathematics allows us to summarize our knowledge into simple formulas. It is why scientists are searching for a theory of everything - a formula that explains reality with 100% accuracy and can make virtually any prediction. What human beings would do with this knowledge is a topic for another thread.

The fact that we can load dice so that they increase the chances of rolling a 6 to almost 100% means that we must know something about dice-rolling. Maybe certainty and knowledge come in degrees rather than in bits. If we had enough information about the dice, the dice-roller, and the environment, we'd be able to predict what the outcome would be.

Notice that if we increase the likely outcomes by giving the dice-roller a 20 sided die (for all you pen & paper RPGers out there) then we increase the amount of information we'd need, increase the difficulty, and lower the probability of a particular outcome. The probability of some outcome is constrained by the possible outcomes in a given process, like dice-rolling with different sided-dice. Rolling a 4-sided dice increases the probability of all the outcomes, and makes it easier to predict the outcome.

Information is the relationship between cause and effect. The more causal relationships we are talking about in any given causal process, the more information in that system. Dice rolling has numerous causal processes involved meaning that it would require for some mind to have access to all of that information to predict the outcome. The more information in that system, the more information we need to make predictions about that system.

We also need to take into account that each particular moment of dice-rolling is different. The conditions of each dice-roll aren't the same. The prediction made about one particular dice roll won't apply to the next because there may be different conditions, like how the dice-roller is holding the dice, or the weather changes, etc. So in each instance, the information changes and we need to update our information in order to make a proper prediction. Because some prediction worked and then doesn't work in the next instance isn't to say that our knowledge is inaccurate, or can't be certain. It is to say that the process is different and so the information we have is not relevant to the current situation. So we would be making a category error in applying information to a situation that it doesn't apply, not that our knowledge can't be accurate.

The world 'appears' deterministic at times at the human scale (e.g billiard balls on a pool table) but this in fact is only an artefact of approximate perception. Is that the origin of the confusion? — Pantagruel

It's not just the world, but my own mind. I have reasons for behaving the way I do, or for the conclusions I come to. That is how reasoning works. You use reasons to support your conclusion. Your reasons are usually observations. Reasoning is causal, and can be predictable when you have access to the information in another person's mind - like when you know how they think because you have the experience of having lived with them for 25 years.

"B knows the initial states" But he cannot know the future with certainty. Some factor that will intervene causing variation. The Neil Armstrong moon landing, a classic example of human intervention, when needed. There is an issue in quality control methods that repeated operations don't produce 'exactly' the same results every cycle. The variations are classified by a level of acceptance/significance.
This problem has its roots in an ideal world of absolute values. Current research denies this.
Re:quantum physics, 'It's not the physics that is strange, but the expectations of the researchers'.