At that point the only apparent mystery that remains is why when you throw the die only 100 times, most of the time each side shows up about 1/6 of the time, and as I described in my previous post that can be explained with statistics, there is no need to invoke any fundamental randomness. — leo

but it is important to see that in some rare cases, even if you pick the initial conditions as randomly as you can, you can still get frequencies that are totally different from the theoretical probability (for instance getting the number three 1000 times in a row even though you have thrown the die in many different ways without knowing the outcome in advance, this is very rare but it can happen). — leo

I just read a very simplified version of the law of large numbers which asserts that as the number of probability experiments increases, the results of the experiment approaches the calculated theoretical probability.

So you're right that "unexpected" outcomes such as 20 threes in a row can occur in a 100 throws of the die. However, as the number of experiments are increased, say to a million throws, the frequency of threes in that million will be approx. 1/6.

At that point the only apparent mystery that remains is why when you throw the die only 100 times, most of the time each side shows up about 1/6 of the time, and as I described in my previous post that can be explained with statistics — leo

Did you mean the law of large numbers?

If yes then that implies the die is behaving randomly. Whence this randomness?

They are guesses and we guess because we are ignorant. — Harry Hindu

You seem to be saying that probability = ignorance but that would imply that there is no such thing as randomness or even chance.

If that's the case then consider:

1. A theoretical probability assumes randomness in its calculations. The theoretical probability for a three is 1/6

2. The die thrown 1 million times will show a three 1/6 of the 1 million throws


2 is exactly as predicted by 1 and 1 assumes randomness.

According to your claim then our ignorance led to the random behavior of the coin? How is this possible? How can my ignorance lead to randomness?

You seem to be saying that probability = ignorance but that would imply that there is no such thing as randomness or even chance.

If that's the case then consider:

1. A theoretical probability assumes randomness in its calculations. The theoretical probability for a three is 1/6

2. The die thrown 1 million times will show a three 1/6 of the 1 million throws


2 is exactly as predicted by 1 and 1 assumes randomness.

According to your claim then our ignorance led to the random behavior of the coin? How is this possible? How can my ignorance lead to randomness? — TheMadFool

You didn't answer those questions I had in my post.

There is such a thing as randomness and chance. They are ideas that stem from our ignorance. Like every other idea, they have causal power. It's just that you are projecting your ignorance/randomness/chance out onto the world where their only existence is in you head as ideas.

There is such a thing as randomness and chance. They are ideas that stem from our ignorance. Like every other idea, they have causal power. It's just that you are projecting your ignorance/randomness/chance out onto the world where their only existence is in you head as ideas. — Harry Hindu

How does my ignorance cause the die to become random?

Separately, I must ask you this:

Are all random and chance events caused by our ignorance?

How does my ignorance cause the die to become random?

Separately, I must ask you this:

Are all random and chance events caused by our ignorance? — TheMadFool

Your ignorance doesn't cause the dice to do anything. Your ignorance causes you to think of the world as probabilities and chances.

The outcome is determined. Your ignorance causes you to not know the outcome. You can only guess at the outcome. Your guess is educated in that you know the possible number of outcomes, but not the actual outcome, hence your educated guess is the probability that exists in your head, not in dice.

If it was all probability, then how is it not probable that you roll a 10 on a six-sided die? What constrains the possible outcomes?

Your ignorance doesn't cause the dice to do anything. Your ignorance causes you to think of the world as probabilities and chances. — Harry Hindu

If it's as you claim, all in my head, how does the die know to come up three 1/6 of the time? Is the die sentient and after finding out I don't have the necessary information to predict, it does everything in its power to ensure that it behaves randomly in such a way as to match my probability predictions?

If it was all probability, then how is it not probable that you roll a 10 on a six-sided die? What constrains the possible outcomes? — Harry Hindu

It can easily be shown the P(10) = 0/6 = 0 = impossible. This has no bearing on why the die is behaving randomly.

So you're right that "unexpected" outcomes such as 20 threes in a row can occur in a 100 throws of the die. However, as the number of experiments are increased, say to a million throws, the frequency of threes in that million will be approx. 1/6. — TheMadFool

It’s possible to throw the die a million times and get three a million times. It’s possible to throw it a gazillion times and get three a gazillion times. Because, again, in a deterministic system the outcome is determined from the initial conditions, so if you always throw the die in exactly the same way you always get the same result.

Now let’s say there are 1 gazillion different initial conditions that yield the outcome three. That means you can throw the die 1 gazillion times in 1 gazillion different ways and always get the outcome three. You can throw it 1 billion gazillion times in 1 gazillion different ways and always get the outcome three.

However if there are N different ways to throw the die (say 6 gazillion ways), and you throw the die once in each way, and the die is perfectly symmetrical, you will indeed get each side with frequency 1/6.

So why is it that most of the time when we throw the die only 100 or 1000 times the frequencies are close to 1/6? The law of large numbers does not explain why if we don’t explain why that law works.

To understand why, consider the following analogy: you have a box in which there are N balls (6 gazillion balls). N/6 balls have the number 1 on them, N/6 balls have the number 2 and so on. If you pick 100 balls arbitrarily, without looking at them, without knowing what their number is in advance, most of the time that you do that about 100/6 balls will have the number 1, about 100/6 balls will have the number 2, and so on. Why?

As I explained earlier, it has to do with combinations, not with randomness. Basically there are more combinations of 100 balls where the 6 numbers show up each with about the same frequency, than there are combinations of 100 balls where the numbers show up with very different frequencies. Try to understand why. If you don’t understand I’ll attempt to find a simple example to make it clear.

so if you always throw the die in exactly the same way you always get the same result. — leo

However if there are N different ways to throw the die (say 6 gazillion ways), and you throw the die once in each way, and the die is perfectly symmetrical, you will indeed get each side with frequency 1/6. — leo

The law of large numbers does not explain why if we don’t explain why that law works. — leo

It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. — Wikipedia

We agree on the point that the outcome depends on and can be predicted by the initial state of the die. The rest of what you said depends on this and we see eye to eye on it.

If you'll allow me to keep things simple and not get into gazillions and combinatorics I think we'll agree on the following:

1. A fair die has 6 sides.

2. For the sake of simplicity assume that each side of the die {1, 2, 3, 4, 5, 6} is an event determined by six initial states {a, b, c, d, e, f} such that a causes outcome 1, b causes outcome 2, c causes outcome 3, d causes outcome 4, e causes outcome 5 and f causes outcome 6.

3. The [i[theoretical probability[/i] for each possible outcome when the die is thrown is 1/6

4. We have to accept that if we knew which initial state obtains we can accurately predict the outcome.

5. Now imagine you throw the die without looking at which initial state the die achieves. You will see the familiar result that each outcome is 1/6 of the total number of times the die is thrown. This concurs with increasing accuracy the greater the number of experiments that are performed.

6. 3 and 5 together imply that the die is behaving randomly

7. We also know that each initial state yields a accurately predictable outcome i.e. each outcome can be known given which initial state the die assumed

8. 6 says the die is random and 7 says the die is not random

9. Somewhere in the chain events, randomness was introduced into the system. The only place possible is at the time you put the die in one of the six initial states and this was random. This makes complete sense when you consider what you said:

Now let’s say there are 1 gazillion different initial conditions that yield the outcome three. That means you can throw the die 1 gazillion times in 1 gazillion different ways and always get the outcome three — leo

I agree for the most part, except:

5. Now imagine you throw the die without looking at which initial state the die achieves. You will see the familiar result that each outcome is 1/6 of the total number of times the die is thrown. This concurs with increasing accuracy the greater the number of experiments that are performed. — TheMadFool

If you don’t look at the initial state, you may pick unwittingly the same initial state every time (or a member of the set of initial states that yield the same outcome), so I don’t agree that we will always see each outcome with 1/6 frequency even with an arbitrarily large number of experiments. In some rare cases the frequencies will be very different, and in order to explain that we have to delve into combinatorics.

And since I don’t agree on this point (5.) I also disagree on your next point (6.), that the die is behaving randomly.

9. Somewhere in the chain events, randomness was introduced into the system. The only place possible is at the time you put the die in one of the six initial states and this was random. This makes complete sense when you consider what you said: — TheMadFool

I do agree that the frequencies of the outcomes are related to the way the initial states are chosen. However I don’t agree that there is a fundamental randomness that is introduced. For instance you can cycle through all the initial states deterministically, and yet you will get the result that each outcome will appear overall with frequency 1/6. So it is not randomness that leads each side of the die to appear 1/6th of the time. In order to explain that, you have to take into account both the symmetries of the die and combinatorics.

The whole reasoning goes as follow: in a deterministic system, for a perfectly symmetrical 6-sided die, it can be shown theoretically (using the symmetries of the die) that each side shows up with probability 1/6 (where probability of an outcome is defined as the number of initial states that lead to this outcome divided by the total number of possible initial states).

Then what remains to explain is why, most of the time, the frequency of each outcome converges towards 1/6 as the number of experiments increases, and yet in some rare cases the frequencies of each outcome are very different even though the number of experiments becomes arbitrarily large. In order to explain that, we have to go into combinatorics.

The probability that this happens becomes increasingly small as the number of experiments becomes increasingly large, but for any finite number of experiments, there are situations where the frequencies of all sides will be very different, even if we pick the initial states arbitrarily without knowing the outcomes in advance. Because for instance, if there are N possible initial states, and you conduct 1000000000000*N experiments, it happens in rare cases that you haven’t gone through all initial states, or that you have picked unwittingly some initial states much more often than some others. And if you don’t take that into account you get the illusion that experimentally we will always see each outcome with frequency 1/6 as we increase the number of experiments, while this isn’t true.

And at the same time combinatorics will also explain while most of the time each outcome shows up with frequency close to 1/6, even though no randomness is introduced at any point. Picking an initial state arbitrarily does not imply that it is picked non-deterministically.

The basic theory of probability and statistics are pretty well established, but when it comes to practical applications there are some problems. Remember when vitamin E was popular for one's health? Then another study showed it had a negative impact on health. Vitamin C is great for this and that, then suddenly it wasn't.

The medical profession does not have a sterling reputation for statistical studies. Sometimes this is due to multiple experiments in which outliers are given undue consideration. Sometimes the experiments are poorly designed.

However, having said this I will tell you that probability theory still leaves me a little uneasy, even though its applications have been largely very successful. Probability waves in QM? Who'd have thunk? :brow:

I'd like to give my own "solution" to the paradox:

A deterministic system can't be random and the die is behaving as if it is random. This implies that a random element was introduced into the system — TheMadFool

It doesn't imply that. A number-generating algorithm simulating 1000 throws can be completely deterministic. Given the algorithm, I could correctly predict every outcome before the simulation is run. For example, that throw 23 would produce a six. Yet all the outcomes taken together would follow a probability distribution.

Perhaps you just want to say that this is not true randomness, it only appears random. If so, the term you could use is pseudorandom.

It doesn't imply that. A number-generating algorithm simulating 1000 throws can be completely deterministic — Andrew M

. Yet all the outcomes taken together would follow a probability distribution — Andrew M

Because...

we can introduce randomness or more accurately pseudo-randomness into a deterministic system.

If you don’t look at the initial state, you may pick unwittingly the same initial state every time (or a member of the set of initial states that yield the same outcome) — leo

This is exactly what bothers me. It should be possible to bias the experiment towards a particular outcome. Yet this doesn't happen and the die behaves in a completely random fashion as is evidenced by the frequency of outcomes in an experiment of large enough number. Why?

However I don’t agree that there is a fundamental randomness that is introduced. — leo

:chin: The evidence for randomness is in the relative frequencies of outcomes in an experiment which perfectly or near-perfectly matches the theoretical probabilities.

The basic theory of probability and statistics are pretty well established, but when it comes to practical applications there are some problems. Remember when vitamin E was popular for one's health? Then another study showed it had a negative impact on health. Vitamin C is great for this and that, then suddenly it wasn't.

The medical profession does not have a sterling reputation for statistical studies. Sometimes this is due to multiple experiments in which outliers are given undue consideration. Sometimes the experiments are poorly designed.

However, having said this I will tell you that probability theory still leaves me a little uneasy, even though its applications have been largely very successful. Probability waves in QM? Who'd have thunk? :brow: — John Gill

:smile: :up: If there's a hidden message in there I didn't see it

Because...

we can introduce randomness or more accurately pseudo-randomness into a deterministic system. — TheMadFool

Yes, so does any puzzle remain for you?

BTW here's a quick-and-dirty example. Suppose my algorithm for 1000 dice throws is to take the first 1000 prime numbers, calculate their square roots, multiply their decimal expansion by 6 and round up to the nearest natural number. The algorithm is deterministic and the sequence is pseudorandom.

The first few throws are:
3, 5, 2, 5, 2, 4, 1, 3

pseudorandom — Andrew M

Correct. I wonder how one differentiates the true random from pesudorandom?

I wonder how one differentiates the true random from pesudorandom? — TheMadFool

From an operational perspective, "true" randomness would come from outside the system of interest (such as from thermal noise or quantum phenomena that are used as inputs to the system), whereas pseudorandomness would be the result of computable processes within the system.

A determinist about the universe would regard them as ultimately the same thing. Randomness and chance would just be terms indicating one's ignorance of the relevant information for making correct predictions. Empirically, one could try to discover the underlying laws or causes that explain apparently random events.

This is exactly what bothers me. It should be possible to bias the experiment towards a particular outcome. Yet this doesn't happen and the die behaves in a completely random fashion as is evidenced by the frequency of outcomes in an experiment of large enough number. Why? — TheMadFool

Because as I keep mentioning, it does happen! It is rare but it does happen, even with an incredibly large number of experiments. There is a non-zero probability that you will throw the die a gazillion times and always get the same number, or never get some number, or get some numbers much more often than the others. The reason you believe it doesn’t happen is that most of the time it doesn’t happen, but sometimes it does happen.

Again this is explained with combinatorics. I’ll give a simpler analogy than earlier. Since there are N initial states, and N/6 initial states lead to outcome one, N/6 initial states lead to outcome two and so on, there are as many initial states that lead to each outcome. And once an initial state is picked, the outcome is already determined (even if we don’t know it in advance).

So a mathematically equivalent analogy of the throw of the die (6-sided and perfectly symmetrical) is to have 6 identical balls inside a box, numbered from one to six. Throwing the die in an arbitrary way without knowing the outcome in advance, corresponds to picking one ball arbitrarily without looking inside the box. Then you write down the outcome and you put the ball back inside the box, and you shake the box so that you have no idea which ball is which when you pick another ball after.

And if you do that many times, it is possible that you will always pick the same ball. As the number of picks increases it becomes rarer and rarer, but no matter how many picks you make it still can happen. Or it can happen that you never pick some ball, or that you pick some balls much more often than some others, or whatever.

So even though this shows that in some rare cases the observed frequencies will be very different from the theoretical probabilities, even with an extremely large number of picks, combinatorics also allow you to show that most of the time the observed frequencies will be close to 1/6 as the number of picks increases.

Because for instance if you make 100 picks, there are 6^100 possible ordered combinations of numbers from one to six (say 1-2-4-3-6-2-... or 3-1-1-1-1-6-...), and mathematically you can show that in most combinations, each number appears with about the same frequency. So most of the time the balls are picked with about the same frequency, most of the time each side of the die appears with about the same frequency, but that doesn’t always happen, because there are some combinations of 100 numbers from one to six in which the numbers have very different frequencies.

Mathematically it’s even possible to calculate how likely it is that after X picks, each number shows up with frequency less than say 2% away from 1/6 (or 1% away or 0.1% or whatever). And if you carry out the calculation (it’s not easy but it’s doable), you will see that there is a non-zero probability that after X picks the frequencies are very different from 1/6, no matter how large X is.

Is there something that still isn’t clear?

Is there something that still isn’t clear? — leo

I'm not saying a given outcome(s) is/are impossible. Perhaps I don't see the relevance of what you're saying to what is a actually bothering me. Kindly read below.

Randomness and chance would just be terms indicating one's ignorance of the relevant information for making correct predictions. — Andrew M

@Harry Hindu look at the part underlined.

I've given it some thought and I think you both are correct but not in the way you think.

Imagine a deterministic system A (a fair die with 6 sides). Once we have all the information on A we can make accurate predictions of how A will evolve. Deterministic systems will have specific outcomes right? There's nothing random in A and so however A evolves, everything in A will show a pattern and there won't be any variation in the pattern.

Please note that patterns are of two types which are:
1. Deterministic patterns. A good example would be gravity - there's a force and that force acts in a predictable manner.

2. Non-deterministic or probabilistic patterns. A die throw is effectively random but any sufficiently large experiment will demonstrate that the outcomes have a pattern viz. that three appears 1/6 of the time, an odd numbered face will appear 3/6 of the time.

Also bear in mind that a deterministic pattern will differ markedly from a non-deterministic/probabilistic pattern. The latter will exhibit multiplicity of outcomes will the former has only one determined outcome.

Imagine now that we lack information i.e. we're ignorant of factors that affect how A will evolve. We assumed A to be deterministic and given that our ignorance has no causal import as far as the system A is concerned, system A should have a deterministic pattern. However, what actually happens is system A now exhibits a non-deterministic/probabilistic pattern.

I will concede that there was a lack of information about system and that is ignorance but that has no causal import on A which should be exhibiting a deterministic pattern because system A is deterministic as we agreed. However, the actual reality when we do experiments we observe non-deterministic/probabilistic patterns.

Perhaps I don't see the relevance of what you're saying to what is a actually bothering me. Kindly read below. — TheMadFool

Yes you aren't seeing the relevance, because it precisely addresses what is bothering you.

I've given it some thought and I think you both are correct but not in the way you think. — TheMadFool

That's the kind of thing I was referring to yesterday, you assume that you know better and that there is something you see that we don't see, while it is the other way around. Your misconception is making you believe that, because you're still not seeing your misconception.

2. Non-deterministic or probabilistic patterns. A die throw is effectively random but any sufficiently large experiment will demonstrate that the outcomes have a pattern viz. that three appears 1/6 of the time, an odd numbered face will appear 3/6 of the time. — TheMadFool

This is wrong. It is false that "any sufficiently large experiment will demonstrate that the outcomes have a pattern viz. that three appears 1/6 of the time". As I keep telling you again and again and again, sometimes no matter how large your experiment is, it doesn't exhibit the pattern you mention. Sometimes you might throw the die 1 billion billion billion billion billion billion billion billion times and always get the same number, or never get some number. It is extremely rare, that's the only reason why you haven't noticed it.

And to explain why most of the time there is the pattern you mention, the answer is, again: combinatorics. There is nothing non-deterministic in combinatorics. It simply says that for a series of X natural numbers between 1 and 6, there are 6^X possible combinations (6^X different series), and that in most of these series, when X is large, each number appears about as frequently as the others, and the larger X is the bigger the percentage of series in which each number has a similar frequency. But no matter how large X is, there are always series where the frequencies are totally different from 1/6. You have to understand that, otherwise you will never get it.

The outcomes that we observe experimentally can be explained fully deterministically by invoking the symmetries of the die and combinatorics, without invoking non-determinism at any point.

Also bear in mind that a deterministic pattern will differ markedly from a non-deterministic/probabilistic pattern. The latter will exhibit multiplicity of outcomes will the former has only one determined outcome. — TheMadFool

This is wrong also, you throw the die in different ways that's why there is a multiplicity of outcomes, otherwise what you're saying would imply that if the die behaves deterministically it would always land on the same side no matter how we throw it, THAT would be the weird thing.

As I keep telling you again and again and again, sometimes no matter how large your experiment is, it doesn't exhibit the pattern you mention. Sometimes you might throw the die 1 billion billion billion billion billion billion billion billion times and always get the same number, or never get some number. It is extremely rare, that's the only reason why you haven't noticed it. — leo

That unexpected events occur isn't an explanation for the issue I raised but...

This is wrong also, you throw the die in different ways that's why there is a multiplicity of outcomes, otherwise what you're saying would imply that if the die behaves deterministically it would always land on the same side no matter how we throw it, THAT would be the weird thing. — leo

I underlined the part that made sense to me. It squares with my explanation. Thanks.

@Andrew M@Harry Hindu

• Deterministic systems can behave probabilistically
• Ignorance or rather the impossibility of knowing was the actual impetus for the development of probability theory

Thanks all

That you can find patterns in a deterministic system doesn’t mean that the deterministic system behaves non-deterministically. That you can find patterns in the frequencies of outcomes when you throw a die doesn’t mean that the die behaves non-deterministically. That you can find patterns in the frequencies of letters in the English language doesn’t mean that English words behave non-deterministically. That you can find patterns in the relative sizes of pizza slices when you cut a pizza doesn’t mean that the pizza behaves non-deterministically...

You are conflating the existence of statistics (which can be expressed as ratios or percentages) with non-deterministic behavior, they aren’t the same thing at all. Both deterministic and non-deterministic systems have statistics, so when you express the statistics of a deterministic system it doesn’t mean that it behaves non-deterministically. It seems to me you believe you have understood while you haven’t really understood.

Imagine a deterministic system A (a fair die with 6 sides). Once we have all the information on A we can make accurate predictions of how A will evolve. Deterministic systems will have specific outcomes right? There's nothing random in A and so however A evolves, everything in A will show a pattern and there won't be any variation in the pattern. — TheMadFool

Just to be clear, deterministic system A is one single die roll. The next die roll would be deterministic system B, and so on.

Just because you know the conditions of A, doesn't necessarily mean you know B. While their might be some causal influence, predicting B is not predicting the same system because you may use your other hand to roll, so the dice will roll in the opposite direction and maybe you roll it with a little less force, and so on. While you may have the formula for the system, you don't have the numbers to plug in for each die roll, unless you know the conditions of each die roll.

Please note that patterns are of two types which are:
1. Deterministic patterns. A good example would be gravity - there's a force and that force acts in a predictable manner.

2. Non-deterministic or probabilistic patterns. A die throw is effectively random but any sufficiently large experiment will demonstrate that the outcomes have a pattern viz. that three appears 1/6 of the time, an odd numbered face will appear 3/6 of the time.

Also bear in mind that a deterministic pattern will differ markedly from a non-deterministic/probabilistic pattern. The latter will exhibit multiplicity of outcomes will the former has only one determined outcome.

Imagine now that we lack information i.e. we're ignorant of factors that affect how A will evolve. We assumed A to be deterministic and given that our ignorance has no causal import as far as the system A is concerned, system A should have a deterministic pattern. However, what actually happens is system A now exhibits a non-deterministic/probabilistic pattern.

I will concede that there was a lack of information about system and that is ignorance but that has no causal import on A which should be exhibiting a deterministic pattern because system A is deterministic as we agreed. However, the actual reality when we do experiments we observe non-deterministic/probabilistic patterns. — TheMadFool

We may know the formula for gravity, which tells us how two massive bodies will interact via gravity, but we still need to know the mass and distance between the two objects in order to predict what will happen over time. We still need to have those conditions plugged into the formula.

So just because we have a formula for a type of system doesn't mean that we can always predict the outcome, because we still need more information because each system is unique, even though they share a common formula/pattern that we refer to as a "system". The "system" is actually a unique state-of-affairs in each moment. It's just that many states-of-affairs share similar qualities, like die rolls. Each state-of-affairs includes hands rolling the same die.

What I think is happening here is that you are confusing the pattern with the state-of-affairs. The pattern is in your mind. The state-of-affairs is out there. Your formula/pattern is only a partial representation of the state-of-affairs. There are other factors that you aren't taking into consideration when determining the outcome of any particular state-of-affairs. If you know ALL the factors, then you can predict the outcome.

Since some of the factors change in the next die roll (but not all of them because we are still using hands to roll the same die), we'd need to know what changed, and by how much, in order to make the prediction for the next state-of-affairs.

This also makes you confuse the two patterns you have. You're conflating knowing one die roll with knowing all of them. How does knowing the formula/pattern for gravity allow you to predict every gravitational event in the universe? Don't you still need to know the mass and distance of the planets or stars in each event?

Rolling dice would be like rolling planets together. How will each interaction evolve? How many times will there be collisions vs. establishing orbits around each other? So you're confusing the pattern in your head as if it were the state-of-affairs itself.

Deterministic systems can behave probabilistically
Ignorance or rather the impossibility of knowing was the actual impetus for the development of probability theory — TheMadFool

:up:

It seems to me you believe you have understood while you haven’t really understood. — leo

Tell me what is it that I didn't understand.

You're conflating knowing one die roll with knowing all of them — Harry Hindu

Correct. Thanks.

Tell me what is it that I didn't understand. — TheMadFool

I’ve done that repeatedly in the previous posts, do I really need to say it again?

You believe that the die behaves non-deterministically, that’s wrong. When you say that it behaves probabilistically that’s what you mean right? That it behaves non-deterministically. But it makes no sense to say that a deterministic system behaves non-deterministically, it’s self-contradictory.

The fact that the outcomes of multiple die throws exhibit often particular frequencies does not imply that the die behaves non-deterministically at any time. Just like the fact that the letters in an English book exhibit often particular frequencies does not imply that English books behave non-deterministically.

Now tell me, what is it that you don’t understand about that?

You believe that the die behaves non-deterministically, that’s wrong. — leo

Well, what is the best way to model a die throw in your view?

1. Probability

2. Determinism

Both right?

Well, what is the best way to model a die throw in your view?

1. Probability

2. Determinism

Both right? — TheMadFool

I think you’re conflating probability and non-determinism. When we say that “there is 1/6 probability that the die will land on three”, do you agree that you interpret it as saying that the die behaves non-deterministically?

It is fine to say that there is 1/6 probability that the die will land on a given side. But this simply expresses our ignorance of the initial conditions when we throw the die, it doesn’t express that the die behaves non-deterministically, and I think that’s where your confusion lies.

To say that the die behaves non-deterministically would be to say that the die doesn’t have a determined trajectory once it is thrown, that it will behave differently even if it is thrown in exactly the same way and even if everything else remains the same, but this isn’t true, or at the very least there is no evidence of that, and the frequencies of the observed outcomes can be explained without invoking any non-determinism.

When the system is deterministic, probabilities refer to incomplete knowledge, but that complete knowledge exists even if we don’t have access to it. Determinism is not incompatible with probabilities, with incomplete knowledge, with ignorance.

But when a system is deterministic, it cannot be non-deterministic, that would be a contradiction. In a fundamentally non-deterministic system, even if we have complete knowledge of it, there are probabilities that remain, for instance the system can behave differently even if it starts from the exact same initial conditions.

Whereas in a fundamentally deterministic system, when you have complete knowledge of it the probabilities disappear, you know exactly which outcome you’re going to get. If after many throws the frequencies of each outcome are about 1/6, you can explain why. If they are totally different from 1/6 you can explain that too. You can explain why in many cases the frequencies are about 1/6. There is no non-determinism.

As evidence that probabilities refer to incomplete knowledge (and not non-determinism), consider the following:

If you don’t know anything about a die, you wouldn’t come up with any probability, you would say that the outcome of a die throw can be anything.

But through thinking, through making use of the symmetries of the die, you can conclude that there are as many ways to throw the die that lead to outcome ‘one’, than there are ways that lead to outcome ‘two’, than there are ways that lead to outcome ‘three’, and so on. This gives you partial knowledge, you know that if you throw the die once in every possible way, each outcome will be realized with frequency 1/6. That’s why we say that without knowledge of the initial conditions, the best we can say is that in all possible ways to throw the die, 1/6th of them lead to outcome ‘one’, 1/6th lead to outcome ‘two’ and so on, or in other words that each outcome has probability 1/6 of being realized.

But if you have also partial knowledge of the initial conditions you can determine the probabilities even more precisely. For instance if you know that in specific ways that you throw the die you never get a ‘six’, then you know that when you throw the die in these ways there is 0% probability to get a ‘six’, and the probabilities of the other outcomes change accordingly. If you know that when you throw the die in a very specific way you always get a ‘four’, then you know that when you throw the die in that way there is 100% probability to get a ‘four’.

And when you have complete knowledge there is no more need to talk of probabilities, you know exactly which outcome you are going to get each time, each time you know there is 100% probability you are going to get a specific outcome.


As another example, through making use of the symmetries of a coin you can conclude that in about 50% of ways to toss a coin you get the outcome ‘heads’ and in about 50% of ways you get ‘tails’. But if you think more carefully you realize that there are also a few ways that lead the coin to land on neither heads nor tails but on its side. So if you take that knowledge into account, you come up with more precise probabilities, you say that there isn’t 50% probability to get ‘heads’ and 50% probability to get ‘tails’ but slightly less, because there is a non-zero probability for the coin to land on its side.

And if you have complete knowledge of the system, of what outcome you get depending on how you toss the coin, then there is no need to talk of probabilities anymore, you have complete knowledge so you know what you’re gonna get, and if you want you can toss the coin so that it always lands on its side.

Whereas in a non-deterministic system, complete knowledge doesn’t allow you to say what outcome you’re gonna get, there are still probabilities that remain, the exact same initial conditions can give rise to different outcomes. Personally I believe that such systems do not exist, that even quantum mechanics can be formulated in a deterministic way, and that probabilities always refer to incomplete knowledge rather than knowledge that doesn’t exist.

I think you’re conflating probability and non-determinism. — leo

What are the following in your view?

1. Probability

2. Determinism

3. Non-determinism