How Nature Preorders Random mathematical Outcomes

Quoting Ergo

Can you ever achieve a jar filled only with only one single color of marbles in it?

Not impossible. but extremely unlikely.

Reply to John
On an infinite timeline with infinite jars it is inevitable.

The chance of winning the Power Ball Lottery is something like one in 175 million, and it still happens.

Quoting Ergo

We will take this into the realm of enormous numbers. What if you filled a trillion jars? Would the outcome be different? A Centillion jars? A Trillion Centillion jars? Would the outcome be different then?

As long as the proportion of marbles is the same and each color has a fair and equal chance of dropping into the jar, then filling more jars would not change anything. The probability of each outcome will remain the same from jar to jar. The only way that probability would change is if the proportion of the marbles changed or if the randomization broke down.

So if we have 5 favorable combinations out of a total of 153,478,146 possible combinations (assuming it is a standard size mason jar holding 115 marbles; just a very rough guess) then we have a 3.257835217E-8 chance of a solid color jar. Hopefully my memory on calculating probability is correct.

Now what happens when we make the jar bigger? Every time we increase the size of the jar the probability gets smaller and smaller.

So if we keep increasing the size of the jar can the probability ever get so small it is 0? It would have to be so big that it could never be filled.

Here is the thing...

Although the math says that infinite jars and infinite chances will "almost certainly" result in a jar filled with only one color of marbles eventually the math that tells us this is not taking into account the inherent nature of the mechanism in question. In order for a jar to be filled with only one color of marbles you would have to violate the complex mathematics governing the chaotic and dynamic processes inside of the vat.

Observe: the same statistical math that leads you to believe that you will eventually "have to" result in a jar filled with only one color of marble (if given infinite tries) also tells us that inside of the dynamic, turbulent, swirling mass of the vat the colors of the marbles will tend to be evenly distributed inside the mass

That means that by the time that the marbles fall out of the funnel located at the bottom of the vat statistically they HAVE to already be distributed by statistical laws. As a result, it would actually defy statistical laws if at any time the statistical distribution of the colored marbles inside of the vat were as such that they would yield an entire jar's worth of marbles of only one single color. It's not just extremely unlikely that such a thing would happen, it is actually a violation of the most fundamental statistical principles.

*

• So the math says "yes" (almost certain) you can/should get a jar filled with only one color of marbles when it is only looking at the end result of marbles falling out of a hole.

• But the same math says "no" (would be in violation) when you take into account the total dynamic processes that are taking place prior to the marbles falling out of the bottom. This is one example of how nature preorders random mathematical outcomes.

Quoting Ergo

it is actually a violation of the most fundamental statistical principles.

We are not really talking about statistic, we are talking about probability. While statistics does incorporate probability mathematics, it itself is a science. The "most fundamental statistical principles" are not actually all that mathematical in nature.

"That means that by the time that the marbles fall out of the funnel located at the bottom of the vat statistically they HAVE to already be distributed by statistical laws"

I didn't realize this as all on a some type of time release, but we can start multiplying probabilities.

Quoting Ergo

As a result, it would actually defy statistical laws if at any time the statistical distribution of the colored marbles inside of the vat were as such that they would yield an entire jar's worth of marbles of only one single color.

Once again we are talking about probability, not statistics; and no it won't. Unless the vat is smaller than the jar, and well vats are not smaller than jars, in fact they are typically very big. Big enough to easily fill a funnel and a jar with a single color.

"Quoting Ergo

This is one example of how nature preorders random mathematical outcomes.

"

You are talking about a human made factory.

Reply to Ergo

Excellent statement about the perplexities of nature. I often wondered myself about the embedded intelligence of nature.

The fact that he does not know the difference between probability and statistics is something of a concern.

I expected your post to simply say, "It has an app for it."

Re your thought experiment, there doesn't seem to be anything random about it, by the way.

It would be practically impossible to predict the exact number of each color that would end up in a given jar, but as long as everything is set up so that marbles are being fed in a steady stream from each color room to the final mix, there's nothing random about that.

Reply to Jeremiah I'm with Ergo on this. The manufacturing process, conveyors, and mixing mechanism are designed to continually produce a consistent mixture of colour. Therefore you will never get a jar of all the same colour marbles unless there is a failure in the mechanism of the factory.

Reply to Metaphysician Undercover

I am sorry, but if you have randomization then you are going to have deviation. There will be a normal distribution which will be based on proportion of the colors, but if we are claiming randomization, then there will be deviation from that distribution. If there is no possibility of deviation then it is not random.

Reply to Jeremiah Well of course it's not random in any absolute sense. The op describes a very specifically, organized mechanical system, therefore the outcome (the filling of the jars) is not random in the sense which you are using "random". The use of the word "random" in the op title is somewhat mischievous, if not downright deceptive, because there is nothing random about the described arrangement.

Personally, I would like to know the source of his information and I would not mind seeing the math, as I feel we may not be getting a fair representation of it here. The way he uses the terminology makes me suspicious.

Quoting Metaphysician Undercover

The op describes a very specifically, organized mechanical system, therefore the outcome (the filling of the jars) is not random in the sense which you are using "random".

Not really; there is no system that could account for this: " They all converge and pour into a large vat where the marbles are mixed together in a torrent of bouncing, clambering mass that flows like fast moving water."

Unless I see something more academic, I am standing by the position: If the out put to the jars is randomized then you can get a full jar of solid colors.

Reply to Jeremiah The outcome isn't randomized though. Ergo uses "random" in the title to throw you off. Nothing in the description of the factory indicates that there is any randomness. Evenly sized, shaped, and weighted marbles are produced, of different colours. They are brought together in even proportions and mixed evenly. The outcome can be nothing other than an even mixture. The system is designed to produce an even mixture.

Reply to Metaphysician Undercover

If the vat is the same size of the jar, but vats are not the same size as jars. Vats are typically much bigger than jars. At any rate, it is sounds more like a word game than one of math.

If you are mixing the vat then there is a probability you'll have pockets of same color marbles.
We can put marbles in a jar and mix them all about and see with our own eyes the distribution. If you are mixing them, then you are randomizing them.

Quoting Ergo

colors of the marbles will tend to be evenly distributed inside the mass

This right here is what we would call a normal distribution, but it is possible to see an event outside the normal distribution.

Weird thread.

[quote=Ergo]That means that by the time that the marbles fall out of the funnel located at the bottom of the vat statistically they HAVE to already be distributed by statistical laws. [/quote]
When you say 'statically xxx' you are saying that some situations/events/etc are more likely than others. Also it's a description not a normative statement. It's weird to hypostatize statistics as if there is some statistical force acting on the marbles.

Although the math says that infinite jars and infinite chances will "almost certainly" result in a jar filled with only one color of marbles eventually the math that tells us this is not taking into account the inherent nature of the mechanism in question.

But does the maths say this?
Your trying to play off two mathematical claims against each other, but if you flesh it out more there is not conflict at all.
1. If there is a the possibility of a jar filling up with a certain color then infinite jars will almost certainly have at least one in which there is only one color.
2. If it's impossible - lets say I set my machine to place one of each color in each jar and it never fails to do this - then even with infinite jars there will not be any with one color in a jar.

The mixing machine is analogous to the mixing of tinctures in a can of paint. If every time you open a can of paint, the colours are well mixed, then the mixing machine has been applied to the parameters necessary for the job. But if sometime you open a can to find a streak of unmixed colour, then the mixing machine hasn't done it's job, and has not been properly designed.

Quoting shmik

But does the maths say this?
Your trying to play off two mathematical claims against each other, but if you flesh it out more there is not conflict at all.
1. If there is a the possibility of a jar filling up with a certain color then infinite jars will almost certainly have at least one in which there is only one color.

The issue here is modelling a deterministic physical system as a stochastic process. If we agree that randomness is an appropriate model, then the probability of finding a jar filled with a single colour would be something like 1 in 4^(n-1) where n is the number of marbles in the jar.

So, if n=101, then you should get a jar full of a single colour every ~1.6x10^60 jars, by which time you will have used up the matter in the universe and created a super-massive black hole.

Reply to Metaphysician Undercover

Look this is simple; take a jar fill it with various colored marbles then shake it around and see if any of the colors are not evenly distributed.

The idea that every single time you mix the marbles you are always going to get an even distribution is just not realistic, and it is not support by the math. A probability distribution always has an element of uncertainty.

Quoting tom

by which time you will have used up the matter in the universe and created a super-massive black hole.

Unless the operators of the factory are environmentalist.

The jar is basically a random sample of of the vat.

So what Ergo is suggesting is that the proportion in the jar will be always be even.

In statistic we would never make an absolute claim like that, because statistic is the science of uncertainty, but we would create a null hypothesis:

Po: P1=P2=P3=P4=P5

Versus an alternative hypothesis

Pa: At least one of the proportions is different.

We would then have to take a jar and measure the results against a null distribution to figure out the probability of the observed results given the null distribution is true. We would then use this p-value or test statistic, to make a conclusion about the hypothesis.

And this is where Ergo's mistake is: He is assuming that given the null is true we will always get an even distribution, because in a fair test after all the math is done we will fail to reject the null; either 90, 95, or 99.95 (typical standards) percent of the time, but there is no always. Yes, we can use the math to approximate a normal distribution but it is called "normal" for a reason.

Here is a simple rundown of the Empirical Rule: http://www.statisticshowto.com/empirical-rule-2/

The process includes an element of uncertainty, and in statistics the conclusion will never be the null is true, it will always be there is strong/weak evidence for (or against) the null (or the alternative which ever may be). And we would make that conclusion based on the probability of the observed results given the null is true.

Statistics does not measure certainty, it measure uncertainty.

Quoting Jeremiah

So what Ergo is suggesting is that the proportion in the jar will be always be even.

I don't think Ergo is suggesting that the proportion will always be perfectly even. The question is, how does the reality of a minor variance in proportion translate, for you, into the possibility of a major variance in proportion? Say you flip a coin 19 times, with 10 heads and 9 tails. Does this suggest to you that if you flip the coin 19,000,000 times you'll end up with 10,000,000 heads and 9,000,000 tails?

Reply to Metaphysician Undercover

I think I identified his mistake. Hypothesis testing will likely support an even distribution. Which to the untrained eye can look like math is proving there will be an even distribution. So I feel he may be misunderstanding that process.

Now we all agree the probability of an all color jar is incredibly low, but that is different then what he was saying.

To those individuals pointing out the difference between Statistics and Probability Distributions and those presenting abstract mathematical thinking—I counter with the fact that "statistically speaking" is a term that most people will understand whether or not they have any fundamental understanding of mathematics or not. The original post is designed in such a way that the common person, laymen or average student (or anyone who watches the news) will be able to engage in the conversation without having to impression upon them the differences between things like Statistical Analysis, Probably Distributions, Discrete and Continuous Data, Discrete Probability, Analog and Digital, Fractals.... so on and so on and so forth.

The reason why I have presented this in this way (in the simplest and most widely understandable terms) is to help the laymen engage in the conversation. Why? Because the common person presumes that when they fill the jar they will get a mix of colored marbles. They presume this because billions of years of evolution has programmed their intuition through a process of natural selection. That being said, how old is the language of math--Babylon maybe? That is but a minuscule fraction of time in comparison. Let us remember that mathematics is a language that human beings have created. We designed it and implement it in our attempts to describe the workings of the physical universe... it is however not in actually the physical universe. It is our attempts to describe and understand it. There is a difference.

In math, if you plug in the wrong values (or if you leave something out) you get a result that appears to be true but in the actual physical universe it ends up being untrue because your math was incomplete. Let me point out the obvious: it is not actually provable that you can eventually end up with a jar filled with only white marbles given the physical conditions that I described. The math that you are using may tell you that it is so, but you cannot actually prove it experimentally in the real and physical world. That means it requires faith on your part to believe that it is true and thus the math you chose to use was complete and you didn't overlook anything. You have to believe that you have accounted for everything when you say “sure... you can end up with a gallon size jar filled with only white marbles if you have infinite tries” even though the average person (who knows nothing about the language of math) knows this to be and untrue statement.

To Summarize: what I am pointing out here is that when you calculate the outcome you may not be putting all of the "math" into the equation. You may not be calculating all the variables. You may just be measuring what is likely to happen as little colored balls fall out of a hole, falsely presuming that the origin of said colored balls is a pure and total random process. In the example above... it is not. The paint analogy presented by Metaphysician Undercover is actually a very good one. If you put a can of paint into a paint mixer you may expect to see a streak of unmixed paint at the top when you opened the lid. But it is not possible for the paint to remain separated after the mixer is done with its work. It does not matter if you have infinite paint cans in infinite universes and infinite tries. If the paint is paint and if the paint mixer does it's job then the paint will be mixed.

This is the way that the actual physical world works and this thread is an attempt to describe how the various physical phenomena of our underlining reality preorders random outcomes.

Quoting Ergo

The paint analogy presented by Metaphysician Undercover is actually a very good one.

No it is not, as we are now talking about chemistry. Marbles are not small enough to fall in that category and behave very differently. I know statistics, maybe someone who knows chemistry can comment on the paint, but I do know marbles are not paint.

I noticed how you didn't try to defend any of your supposed statistical "laws". Could you tell us what those laws are?

Quoting Ergo

You have to believe that you have accounted for everything when you say “sure... you can end up with a gallon size jar filled with only white marbles if you have infinite tries” even though the average person (who knows nothing about the language of math) knows this to be and untrue statement.

Perhaps you could extend some sympathy to those poor souls, lacking your deep statistical intuition, who are condemned to think abstractly about such situations? A simplified example might be useful:

Say your jars could only hold 3 marbles. How often could you expect to find a jar with only one colour in it, and why?

Quoting Ergo

Let me point out the obvious: it is not actually provable that you can eventually end up with a jar filled with only white marbles given the physical conditions that I described.

There is reliance on what is called "the principle of plenitude". This principle states that if something is possible, then if given an infinite amount of time, that possibility will necessarily be actualized. That is why, even if it is provable that a jar of only white marbles is possible, it is still not provable that this will actually occur, because it assumes an infinite amount of time. So we have an unprovable relationship between the finite and the infinite, which is simply assumed here. Each time we fill a jar, we have a definite (finite) occurrence. We are assuming that "time" is such that if we continue with finite occurrences, time will give us the capacity for infinite repetition.

So we must make a particular assumption about the nature of time in order to support the principle of plenitude. There is a relation between time and numbers, such that numbers are an order, two comes "after" one, and three comes "after" two. We assume that one could theoretically keep counting forever and therefore numbers are infinite. But we could use the principle of plenitude against itself. We could say that it is possible that time will end. Therefore if given an infinite amount of time, it is necessary that we will reach the end of time, and this negates the infinite amount of time, in a paradoxical way.

But we can put the principle of plenitude aside because it only comes into play after the possibility is established. I believe that it is Ergo's claim that a jar of all one colour is not even possible under the described circumstances. And Jeremiah claims that it is possible. I have a question for Jeremiah. If slight variances in the mixture, from one jar to another are observable, what leads you to the conclusion that a jar of all one colour is possible?

Quoting Metaphysician Undercover

If slight variances in the mixture, from one jar to another are observable, what leads you to the conclusion that a jar of all one colour is possible?

And where did you establish that only slight variations can occur over an infinite number of jars? If we say something can happen outside normal distribution then we are saying an occurrence that is not a slight variation can occur. I already went over this.

Quoting Jeremiah

And this is where Ergo's mistake is: He is assuming that given the null is true we will always get an even distribution [This does not mean exactly even.], because in a fair test after all the math is done we will fail to reject the null; either 90, 95, or 99.95 (typical standards) percent of the time, but there is no always. Yes, we can use the math to approximate a normal distribution but it is called "normal" for a reason.

Here is a simple rundown of the Empirical Rule: http://www.statisticshowto.com/empirical-rule-2/

Quoting Jeremiah

And where did you establish that only slight variations can occur over an infinite number of jars? If we say something can happen outside normal distribution then we are saying an occurrence that is not a slight variation can occur. I already went over this.

What would constitute something happening "outside the normal distribution"?

Reply to tom

http://www.statisticshowto.com/empirical-rule-2/

The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The empirical rule can be broken down into three parts:

68% of data falls within the first standard deviation from the mean.
95% fall within two standard deviations.
99.7% fall within three standard deviations.

Something that happens outside the third standard deviation.

Quoting Jeremiah

Something that happens outside the third standard deviation.

I gave a formula earlier for the density of occurrences of jars of a single colour. By definition, when this happens, it is well outside many standard deviations for moderate sized jars.

The normal distribution does not end at 3 standard deviations.

Reply to tom

Technically it is a bell curve, so it really does not have an end. My point being due to the low probability you will likely fail to reject the null and it will look like the math is proving an even distribution of the marbles. So I think the math is being misunderstood to mean you will always be within 3 SDs, when that is just not true.

I have to also point out, we are all just assuming there will be roughly an even distribution of the marbles in the jar, but this is not something that has been proven. The only way to get reliable answers would be to actually do the experiment.

Quoting Ergo

You have to believe that you have accounted for everything when you say “sure... you can end up with a gallon size jar filled with only white marbles if you have infinite tries”

This right here vs. this:

"That means that by the time that the marbles fall out of the funnel located at the bottom of the vat statistically they HAVE to already be distributed by statistical laws -Ergo"

Have you accounted for everything? Did your Godly brain uncover all confounding variables? I am sorry, but until you actually run the experiment you don't really know how they will distribute.

You cannot prove they will be distributed on the "statistical law" alone. In fact you are violating a few rules of statistics by making your claim without any data to back it up.

I'll agree with the point: that there may be something unconsidered which will prevent a jar of all colors (which would mean we don't have randomization); however, that also applies to the assumption they will be evenly distributed.

The truth is we are working a hypothetical, and what is needed to get real answers is to actually do the experiment.

Quoting Jeremiah

I'll agree with the point there may be something unconsidered which will prevent a jar of all colors, however that also applies to the assumption they will be evenly disturbed.

The only assumption is that the marbles are well-mixed.

Reply to tom

He is also making an assumption about even distribution. I am not sure if that is what you are referring to with "well-mixed".

Quoting Ergo

colors of the marbles will tend to be evenly distributed inside the mass

To be honest, I can't believe I over looked that detail, guess I was not paying close enough attention. We don't actually know if the marbles will be evenly distributed.

Quoting Jeremiah

And where did you establish that only slight variations can occur over an infinite number of jars?

As I said, we should drop the possibility of an infinite number of jars, because this is an appeal to the principle of plenitude, which in this case, establishes an unreal relation between the finite and the infinite. It assumes the possibility of an infinite number of jars, without demonstrating how this is a real possibility. I believe that due to the nature of time, and physical existence, this is not a real possibility. This is the weakness of Ergo's op, it asks about infinite jars. When you ask unreal questions you get unreal answers.

Quoting Jeremiah

If we say something can happen outside normal distribution then we are saying an occurrence that is not a slight variation can occur.

A slight variation in the proportions of coloured marbles is the normal distribution, just like when you flip coins, a slight variation in the number of heads and number of tails, is the normal. You appear to be arguing that the existence of such a slight variation indicates the possibility of a large variation, if given enough repetitions. On what principle do you argue this? Consider the example I gave already:

"Say you flip a coin 19 times, with 10 heads and 9 tails. Does this suggest to you that if you flip the coin 19,000,000 times you'll end up with 10,000,000 heads and 9,000,000 tails?"

The question is, how do you get from the continued observance of small variations, which would be your premise, that there is not exactly the same number of each coloured marble in each jar, to the conclusion that a large variation (one much larger than any observed variation), is possible? What would be the premise which you would apply to give you this conclusion?

Math will allow us to calculate the probability of it happening. Does this prove it will happen? Not necessarily, but it does suggest it is a possibility, even if it is a very slim one. And the math is making a far more convincing argument than your words.

One of the reasons I study math is so I can philosophize in mathematics as well as words.

I kind of think this discussion is at its end, Ergo's "hypothesis" has been shown to have many flaws.

Quoting Jeremiah

No it is not, as we are now talking about chemistry. Marbles are not small enough to fall in that category and behave very differently.

You are fundamentally incorrect. The laws of physics do not change between a can of paint and a gigantic clambering vat of swirling marbles. The change only occurs in your mind and the unfinished math that you employ as a result reflects an unfinished answer.


Quoting Jeremiah

I kind of think this discussion is at its end, Ergo's "hypothesis" has been shown to have many flaws.

It is my thinking that this particular discussion about randomness is among the most important debates in science, physics, mathematics and philosophy. I must also now point out that you have not actually presented any evidence to show that my original hypothesis has many flaws. You only concluded, that it does, offering no real world representations to support you opinion only more unfinished math.

But in the end, that's how these things tend to unfold. Challenging a principle often yields the same result as challenging someones faith or strongly held opinion, because some principles are simply opinions taken on faith and are not provable experimentally.

Quoting Ergo

Observe: the same statistical math that leads you to believe that you will eventually "have to" result in a jar filled with only one color of marble (if given infinite tries) also tells us that inside of the dynamic, turbulent, swirling mass of the vat the colors of the marbles will tend to be evenly distributed inside the mass

That means that by the time that the marbles fall out of the funnel located at the bottom of the vat statistically they HAVE to already be distributed by statistical laws. As a result, it would actually defy statistical laws if at any time the statistical distribution of the colored marbles inside of the vat were as such that they would yield an entire jar's worth of marbles of only one single color. It's not just extremely unlikely that such a thing would happen, it is actually a violation of the most fundamental statistical principles.

Isn't there an important difference between "tending to be evenly distributed" and "necessarily being evenly distributed"?

It sounds as if you've taken the general tendency and turned it into a universal law of nature admitting of no exceptions. I'm not sure that's how "statistical laws" are supposed to work.

Quoting Ergo

The laws of physics do not change between a can of paint and a gigantic clambering vat of swirling marbles.

Yes, they do. Liquids behave differently than solid marbles.

Saturation is the point at which a solution of a substance can dissolve no more of that substance. This point of maximum concentration, the saturation point, depends on the temperature of the liquid as well as the chemical nature of the substances involved. If a change in conditions (e.g. cooling) means that the concentration is higher than the saturation point, the solution has become 'supersaturated'.
In organic chemistry, a saturated chemical compound has no double bond or triple bond or ring. In saturated hydrocarbons, every carbon atom is attached to two hydrogen atoms, except those at the ends of the chain, which have three hydrogen atoms.
In biochemistry, the term saturation refers to the fraction of total protein binding sites that are occupied at any given time. Applies to enzymes, and molecules like haemoglobin.
In organometallic chemistry, an unsaturated complex has fewer than 18 valence electrons and thus is susceptible to oxidative addition or coordination of an additional ligand. Unsaturation is characteristic of many catalysts because it is usually a requirement for substrate activation.

https://simple.wikipedia.org/wiki/Saturation_(chemistry)

Quoting Ergo

It is my thinking that this particular discussion about randomness is among the most important debates in science, physics, mathematics and philosophy.

References? If it is such an important debate surely you can manage that.

Quoting Ergo

I must also now point out that you have not actually presented any evidence to show that my original hypothesis has many flaws. You only concluded, that it does, offering no real world representations to support you opinion only more unfinished math.

Let me get this straight you are now using unfalsifiability to justify your claim? You do realize that a hypothesis must be falsifiable in order for it to actually be a valid hypothesis, right? It is becoming more and more clear that you do not know much about science or statistics.

I think this is an ulterior motive behind this thread.