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*** Mod's note: Split from the 'racism' thread ***

Viking, i think i see where you are missing it.

Let me ask you a small question. There is a certain (hypothetical) country in which 80 per cent of the children born are girls and only 20 per cent are boys. You meet a woman in that country who is pregnant and she asks you "what is the probability that my baby will be a boy?" What would be your answer to her question?

(Anyone can try to answer this one - it's not a trick question and it's not off-topic as i will show in a moment).

As stated, it's really impossible to answer the question.

If the cause is that people abort male foetuses, then the probability will be approximately 50%. If the cause is something in the air or diet that somehow impedes the growth of male foetuses, then you'd have to figure if you can change those factors -- maybe move to another country and change one's diet.

Bottom line: without knowing more, it's really impossible to answer since living in that country cannot be considered a causal factor with the evidence provided and with everything else we know.

Edited by softwareNerd
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Viking, i think i see where you are missing it.

Let me ask you a small question. There is a certain (hypothetical) country in which 80 per cent of the children born are girls and only 20 per cent are boys. You meet a woman in that country who is pregnant and she asks you "what is the probability that my baby will be a boy?" What would be your answer to her question?

(Anyone can try to answer this one - it's not a trick question and it's not off-topic as i will show in a moment).

I'll bite...I would say the best probability I could give her is 20%.

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Viking, i think i see where you are missing it.

Let me ask you a small question. There is a certain (hypothetical) country in which 80 per cent of the children born are girls and only 20 per cent are boys. You meet a woman in that country who is pregnant and she asks you "what is the probability that my baby will be a boy?" What would be your answer to her question?

(Anyone can try to answer this one - it's not a trick question and it's not off-topic as i will show in a moment).

Go on then..i shall be brave/foolish......20%

The above is based on the following assumptions:

Each man has 4 wives

All women have, on average, the same number of children

Without the above assumptions....erm...I'm not sure...

Edited by SteveCook
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The only right answer you can give is 50%, there is equal probability that it can be a girl or a boy. The stats in the population have to be ignored, logically, and the closest reason for this is softwarenerd's last sentence:

"Bottom line: without knowing more, it's really impossible to answer since living in that country cannot be considered a causal factor with the evidence provided and with everything else we know." [except for the impossible bit].

Can i now go back to the racism debate to put this in context?...

Edited by blackdiamond
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The only right answer you can give is 50%, there is equal probability that it can be a girl or a boy. The stats in the population have to be ignored, logically, and the closest reason for this is softwarenerd's last sentence:

"Bottom line: without knowing more, it's really impossible to answer since living in that country cannot be considered a causal factor with the evidence provided and with everything else we know." [except for the impossible bit].

Can i now go back to the racism debate to put this in context?...

That is wrong, because it violates the conditions you gave:

there is a certain (hypothetical) country in which 80 per cent of the children born are girls and only 20 per cent are boys.
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I'd like to add further context to this probability discussion, which is that there are three distinct concepts of "probability". They do unify at a certain level, but it is essential that one not be misled into thinking that they are interchangeable. First, there is the traditional notion regarding level of evidence -- relating "probable" and "proof". If there is no evidence regarding the child's sex (no ultrasound or other such tests), there is no proof, and the probability of the child being a boy is 0. There is also the strictly mathematical, combinatorial exercise that we learned in high school, where the combinatorial probability of rolling a 6 is 1/6 given a 6-sided die. In this case the answer is 25% (the possibilities being boy, girl, both, neither). This number is computable only given a mathematical model, so I'm treating sex as being two properties which are independently allowed. We know that it's possible to have both, so my model explains that (which the 50% model doesn't explain). An alternative model might only allow boy, girl and hemaphrodite (so the answer would be 33%). I don't know if "none of the above" is actually a physical possibility, but it doesn't matter, because combinatoric probability is only about mathematical distribution, not real-world facts. And finally, there is observed frequency, in which case 20% would be the answer.

When the observational probability is seriously at odds with the combinatorial probability, you have proof-probability evidence that you are missing some fact in your model. Observational probability is utterly meaningless by itself, and the expression "all things being equal" is one of the greatest intellectual crimes of the past 100 years. We have strong evidence that the two-outcome model of sex is false. Then we need to understand why there are so few examples of hemaphrodites, and this also leads us to inquire into whether there is the fourth possibility (a question that I haven't looked into yet, because I gotta split -- anyone wanna post the answer?).

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David,

I'm thinking the answer "0%" would freak out any woman who asks you about the gender of her baby! :lol: . But your post is food for thought: needs some digestion.

Viking,

i don't believe i've violated the condition in the question. "Stats in the population" do include birth rates.

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I'm thinking the answer "0%" would freak out any woman who asks you about the gender of her baby! :lol:
Maybe, but a woman who would ask such a question would be strange, anyway.

A bit of further research indicates that there are at least 5 categories, using genitalia-based criteria. Hence 20% would be a model-correct answer, for that 5-valued model. Proof of something about that country, but I'm not entirely sure what.

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Maybe, but a woman who would ask such a question would be strange, anyway.

A bit of further research indicates that there are at least 5 categories, using genitalia-based criteria. Hence 20% would be a model-correct answer, for that 5-valued model. Proof of something about that country, but I'm not entirely sure what.

20 percent would be the basic correct answer for any country, under that 5-valued genitalia-based model, so it would not prove anything about "that" country in my question.

However, practically, when a person asks you what their probability of having a male child would be, i think they all assume a 2-valued model as your working boundary if you are going to take a genitalia-based criteria. The other possibilities are so existentially rare as to justifiably be considered completely improbable.

Or, to put it another way, when someone has a child, everyone only asks "is it a boy or a girl?" and not ...(i can't even construct the alternative 5-valued question!)

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We have strong evidence that the two-outcome model of sex is false. Then we need to understand why there are so few examples of hemaphrodites, and this also leads us to inquire into whether there is the fourth possibility (a question that I haven't looked into yet, because I gotta split -- anyone wanna post the answer?).

Are there documented cases of humans being born without either male or female genitalia? That would probably be a fourth case. This is not the kind of thing that I wish to Google on my work station.

Perhaps we could also include ludicrous possibilities such as two variants of bi-curious, emotionally transgendered individuals.

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20 percent would be the basic correct answer for any country, under that 5-valued genitalia-based model, so it would not prove anything about "that" country in my question.
Presumably, since the model only has 5 possible outcomes, with no dependencies determined by country. Remember that a combinatoric model is not dependent on fact, it's only dependent on math. Let's change the model a bit. If you have a model with two groups of objects A and B, A contains 2 things (a1, a2) and B has 3 things (b1, b2, b3) and a rule that says "Select exactly one from A and one from B", then you get 6 possibilities and a .166 probability of any particular outcome. Now complicate the model by saying "In hypothetical country Letswana, select exactly one from A and one from B as long as the result is not {a1,b1}": we have a .2 probability. As a mathematical exercise, you just have to set the parameters based on whatever floats your boat.

Personally, I think just looking at combinatorics or frequency distribution is uninformative, except as a way of clarifying whether the scientific model might be missing a causal factor. Thus a significant difference between combinatoric probability and observed frequency tells you that you don't know what causes the event being tested for, and you need to inquire into why so many girls are being born in this one country. That would reveal the evil secret poisong plot in the capital city, which affects fetuses concieved in women drinking the water of the capital city.

However, practically, when a person asks you what their probability of having a male child would be, i think they all assume a 2-valued model as your working boundary if you are going to take a genitalia-based criteria. The other possibilities are so existentially rare as to justifiably be considered completely improbable.
How doesn't that undermine what I understand to be your claim? The probability of a boy is so existentially rare as to justifiably be considered to be less than 50%, in fact, about 20% (the observed frequency).
Or, to put it another way, when someone has a child, everyone only asks "is it a boy or a girl?" and not ...(i can't even construct the alternative 5-valued question!)
I don't blame people for making errors of modelling based on ignorance, unless they're professionals in that area of inquiry. But then most people don't know how to compute probabilities based on high school combinatorics. You don't even need to get bogged down in the details of hermaphrodism and pseudo-hermaphrodism to find problems with the 50% model. Some women favor production of females -- don't understand why, but it's there. Then such a woman could have a 20% chance of a boy or even a 0% chance, depending on that personal body factor. If you have a bad model of baby-sex or race, you will, of course, be able to compute bad numbers.
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I think that "probability of an event" is used as a shortcut for "probability of predicting the event correctly". I think this is fine, but it raises the question of whether one should speak of probability from a universal context of knowledge or from the context of an individual observer. i.e. is it "probability of predicting the event, given the best of human knowledge", or is it "probability of predicting the event given a particular person's knowledge".

Using BlackDiamond's example, suppose the woman who asked the question about baby's sex just had an ultra-sound where the doctor told her that her baby is male.

  • The probability that any live birth will be male is 100% (or pretty close)
  • OTOH, if one does not know this, the probability that one will guess right is not 100%.

Suppose the woman asks the question this way: "I know the sex. The doctor told me he was certain. What is the probability that it is male?" The answer "50%" or "20%" no longer makes sense. In the real world, when one uses the term "probability of an event", one is speaking of the "probability of predicting that event, given all relevant human knowledge". So, one might respond, saying "you already know for certain, but the probability that I can guess it right is XX%".

The alternate view is that probability is not a universal measure, but a subjective measure, and that one cannot ask about the probability of an event without taking into account the particular knowledge (and errors) of the observer. So, with this view, in the example above, one might say "Well, Ma'am, for you the probability that it is male is either 100% or 0%; but for me, the probability is XX%".

I think this latter way is incorrect. I think one should always distinguish between the two types of usage.

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How doesn't that undermine what I understand to be your claim? The probability of a boy is so existentially rare as to justifiably be considered to be less than 50%, in fact, about 20% (the observed frequency).

It probably does undermine my claim a little. I was trying to cross the bridge from math to "reality" (hence my opening that statement with the word 'practically') and i thought that the fact that these other genital possibilities are so extremely rare, not just in that country but everywhere else in the world, and at any time in the history of the world, would justify eliminating them or taking them as practically non-existent ("existentially"). But thinking about it more, I'm not so sure ... i'm also begining to consider that this is probably as useless as the observed frequency for "practical" purposes.

But wait a minute! Another thought crossing my head right now ... what about the fact that in almost every (non-hypothetical) country in the world the observed frequency for males and females (births) is approximately 50%, and this coincides with the 2-modeled prediction for a specific birth? Doesn't that mean anything? I'm also thinking that the observed frequency when you toss a coin a huge number of times also approximately coincides with the combinatorial probability. Doesn't this make the combinatorial probability a bit useful? And doesn't this in a way justify my reasons for "practically" ignoring the five-modeled formula?

(Now, this is what i call "thinking out loud"! I'll come back with further thoughts if i see an error in my idea ... a friend is calling me to go play some pool!) Please feel free to correct the idea in my last paragraph (like i even need to say it. :wub:)

Have a nice weekend, gentlemen (and ladies? what's their observed frequency on these forums, btw? do we all assume a new person on the forum is a man - until proven innocent? :))

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Suppose the woman asks the question this way: "I know the sex. The doctor told me he was certain. What is the probability that it is male?" The answer "50%" or "20%" no longer makes sense. In the real world, when one uses the term "probability of an event", one is speaking of the "probability of predicting that event, given all relevant human knowledge". So, one might respond, saying "you already know for certain, but the probability that I can guess it right is XX%".

The alternate view is that probability is not a universal measure, but a subjective measure, and that one cannot ask about the probability of an event without taking into account the particular knowledge (and errors) of the observer. So, with this view, in the example above, one might say "Well, Ma'am, for you the probability that it is male is either 100% or 0%; but for me, the probability is XX%".

I think this latter way is incorrect. I think one should always distinguish between the two types of usage.

I think i would still say the answer is 50 (or 20 if my post above is wrong). The thing is, if she KNOWS the sex, then "the probability that she can guess it right" does not apply to her. The situation is the same as someone joining this forum and asking everyone on the forum, "what is the probability that i am male?". I will say to him/her that the probability is 50% (or 20), of course; since he KNOWS what he is, the question cannot usefully be directed to him as well. So, maybe I can revise your interpretation of probability of an event as: "the probability that one who doesn't know the answer can guess correctly." Which is a bit tautologous because guessing (or predicting) only applies to those who don't know!

This, of course, doesn't make it a subjective measure because the calculation can be done by anyone, including the one who KNOWS.

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David,

I'm thinking the answer "0%" would freak out any woman who asks you about the gender of her baby! :) . But your post is food for thought: needs some digestion.

Viking,

i don't believe i've violated the condition in the question. "Stats in the population" do include birth rates.

Please re-read what I said, and take note of the part of your quote that I put in bold. You specifically said BORN, not population stats. If you had said population stats, then you would be right, and naturally I would have said 50%. But, since you said "born", you are dead wrong.

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Another thought crossing my head right now ... what about the fact that in almost every (non-hypothetical) country in the world the observed frequency for males and females (births) is approximately 50%, and this coincides with the 2-modeled prediction for a specific birth? Doesn't that mean anything? I'm also thinking that the observed frequency when you toss a coin a huge number of times also approximately coincides with the combinatorial probability. Doesn't this make the combinatorial probability a bit useful?
Right, so plain combinatorics tells you that boys should be 20% of the population and girls should be 20%, and merms, ferms and hermaphrodites should be the majority -- combinatoric probability. The fact is that the predicted 60% categories are under 1% of the population -- frequency probability. This means that your initial hypothesis of 5 equal cases was wrong, and you should inquire into what causes sex differentiation in humans, which leads you to discover that the developing body can fail to respond to androgen, which is in some cases due to mutation of the gene AR on the X chromosome. You could fall in the statistical margins in terms of having the mutation if you inherited it from your mother, or if it is a rare new mutation. Dunno how or how often that happens. Anyhow, now you can cook up a new model that better matches the facts. Then if the woman asks this question, you can whip out a questionaire that may help you better compute the outcome for this particular woman.

Since AIS is rare, we could just ignore it as a possibility and live with some minor numeric anomalies. The underlying logic is still applicable to the 20% boy problem, which again would be caused by something -- what? Something in the water, I'm guessing, thus the mother actually had to drink the water, and maybe she had to eat lake fish within a week of conception. There you have it: if you know the factors that cause events, you can predict the events.

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  • 1 month later...

Just had some interesting thought about probability, statistics in relation to laws of nature that I would like to put down.

first, some background thoughts: I am currently taking a course in statistics, after studying intro to probability last semester in my university.

In statistics we study the way to process a series of measurements or observations into a model of probability to be used for predicting future results of the same kind.

There are two important concepts: population, and series of sampling (not sure if I am using the correct terms in English here, since I'm translating my knowledge from Hebrew, so take that into account).

We seek knowledge about the population, but in reality use a smaller series of sampling from a part of the population to induce a probability model to teach us about the entire population.

The population may be finite (like the collection of individuals living in California at the moment, or the collection of numbers (1,2,3,4,5,6). ), and infinite, like the collection of individuals that were ever born, and will be born in a certain county.

The probability of the population is the true probability, and the one provided to us by statistical processing of the series of sampling is probability with a certain error that can be estimated.

Anyway, I was thinking about how we verify laws of nature. I was thinking about F=ma (second Newton law), a law that was thought to be correct for many years, and was validated in endless experiments, in a certain range of speeds.

The way I see it, the law actually says that the relation F=ma is true for all forces, masses and accelerations possible. (Before it was shown to be false in high speeds).

But in fact, experiments have only been done (before the law was proven wrong) for a small part of that "population" of possible combination of forces and masses. The comparison I made is that the "sampling" was not enough to teach about the "population" in this case, when the population is the endless series of possible combinations (F,m), and the sampling is the series of results from experiments.

When can we tell that we have conducted enough experiments so that a law is 100% correct?

What is the method that we must choose the sampling in order to learn about the "population"? How can we know that if the law is true for (F=100,m=20) that it will also be true for (F=100,m=21)?

The possible singularity may exist in some number in the middle (due to some unknown factor), and not just when the speed approaches the speed of light. So how should one choose the series of experiments to cover all bases, if that is possible at all?

It is impossible to go over all the numbers, since the population is infinite in this case (series of all possible (F,m) ), so how can we ever know that this is true by using samples of it?

Note: There is a difference between the concept of probability of population and actually having correct knowledge about the population. In statistics we seek knowledge of true probability, while in my F=ma example I was asking about true knowledge of the population, and not probability of it.

Note 2: this post might sounds a bit confusing, because I am using some concepts without describing them first. So if something isn't clear, and you have something to contribute, just ask. Thanks.

Edited by ifatart
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The general question regarding scientific laws and their validation is, in broad strokes, addressed in OPAR ch. 5, and I assume familiarity with that. As you know, an inference from a sample to properties of the population is mathematically valid only if all members of the population have an equal chance of being included in the sample. That was manifestly not the case for Newtonian mechanics. What this means is that multiple conclusions remained valid. Combined with the law of gravity, it was known from the early 19th c. that the classical model of planetary motion was false, i.e. there existed observational reason to reject the conclusion, meaning that the conclusion was not certain.

When can we tell that we have conducted enough experiments so that a law is 100% correct?
It is never about conducting enough experiments: it is always about dispelling reason to doubt. If you have a faulty instrument as your detector, a million experiments will still give you crap. Whenever you extrapolate from obversation to non-analogous circumstances, you are conjecturing, and untested conjectures are a sign that your law may not be valid, insofar as it is based on an unvalidated assumption (which might be something like "God designed the universe to be mathematically elegant"). Notice that the Newtonian account of gravity was, for all intents and purposes, completely untested except for a tiny range of possibilities, and when a scientific law purported to hold for all distances and masses is tested only for a stunningly restricted range of values, the conclusion cannot be said to be certain.
It is impossible to go over all the numbers, since the population is infinite in this case (series of all possible (F,m) ), so how can we ever know that this is true by using samples of it?
You mean unbounded (or else you're simply wrong). We don't really know that that is so. Natural law is not about "the numbers", it's about existents. If there is no existent s.t. F=100, m=21 (or, F=1010000000000000000) then you don't need to worry about testing that case, since it isn't a case.
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When can we tell that we have conducted enough experiments so that a law is 100% correct?

Hi Ifat,

David articulated this well. Although empirical observation is important, you can never validate a hypothesis because you observed a certain number of empirical observations. Instead, you need to identify a satisfying causal explanation, dispell all reasonable forms of doubt and be able to properly integrate the conclusion with your entire present context of knowledge.

What is the method that we must choose the sampling in order to learn about the "population"? How can we know that if the law is true for (F=100,m=20) that it will also be true for (F=100,m=21)?

The possible singularity may exist in some number in the middle (due to some unknown factor), and not just when the speed approaches the speed of light. So how should one choose the series of experiments to cover all bases, if that is possible at all?

It is impossible to go over all the numbers, since the population is infinite in this case (series of all possible (F,m) ), so how can we ever know that this is true by using samples of it?

All of these questions, including the first one, essentially pivot around the issue of induction. Dr. Leonard Peikoff has an excellent series of lectures entitled Induction in Physics and Philosophy. He is also currently writing a book on this very topic along with David Harriman. I recommend getting excited over the release of this book!

Of course, you might also just want to search for threads on questions of induction, as it might help further your understanding.

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Statistics has two good uses: (1) predicting events in the absence of complete knowledge, and (2) as a tool for finding causal relationships. On its own, statistics can only establish correlation, not causation. To establish causation requires further thinking about the cause-and-effect mechanism at work. Also, it requires understanding all relevant causes, and being able to explain the relevant examples.

Personally, I think just looking at combinatorics or frequency distribution is uninformative, except as a way of clarifying whether the scientific model might be missing a causal factor. Thus a significant difference between combinatoric probability and observed frequency tells you that you don't know what causes the event being tested for, and you need to inquire into why so many girls are being born in this one country.

Yes, that is why statistics is a good tool for the researcher. But it takes further work than statistics to establish causality. Statistics on its own can only say that there exists some kind of relationship among several factors. Specifically, they are correlated. There may or may not be a causal relationship. Obviously, the stronger the correlation, the greater the likelihood that there is some kind of causal mechanism at work. The virtue of statistics is that it provides a precise measure of the degree of correlation. It gives precision (a measure of the degree of correlation) to something that remains unknown (the exact causal relationship among phenomonena).

In the real world, when one uses the term "probability of an event", one is speaking of the "probability of predicting that event, given all relevant human knowledge".

This is also what statistics does. It is a tool for prediction, and quite a useful one at that. It is a tool for prediction when all of the causal factors are not known or they would be too costly/difficult to measure if they were known. For example, a drug manufacturer may establish that a certain cancer drug successfully treats cancer in 80% of patients. Until further (costly) research is done on the causal mechanism, that is all he can say. Later, he learns through further research that the 20% of patients it did not work on actually have a slightly different form of cancer, or that these patients have a common genetic mutation that makes the drug ineffective, etc.

Even without having complete knowledge of the causal mechanism, the drug manufacturer can confidently advise doctors to prescribe the drug now, knowing that approximately 80% of the patients they treat are likely to benefit. Of course, this number is bounded by a confidence interval, which is a statistical way of stating how certain one can be that the correlation is 80% and not 90% or 70%. Statistics provides a mathematical way of knowing how many experiments (individual samples) are necessary to establish a correlation with a particular degree of certainty.

There you have it: if you know the factors that cause events, you can predict the events.

Full knowledge of how causation works, including all relevant causal factors, is necessary to make certain predictions. It is not necessary to do statistical tests over and over again. For example, if one understands the laws of motion, and the nature of billiard balls, it will only take a few shots at a pool table to understand the motion of billiard balls, and to be able to predict how billiard balls will move given a variety of shots.

In another example, in economics one can state that the law of supply and demand is an ironclad principle. One doesn't establish this by doing a statistical study and observing that in 94% of instances, the quantity of goods purchased declined when the price rose. One only needs a sufficient number of examples (not an inordinate number) coupled with thinking about the nature of the price mechanism in a free market to establish the validity of the law of supply and demand.

As for the 94% statistic, further work explains those 6% of instances where the law of supply and demand does not seem to work. For example, economists talk about an income effect. If a person's income simultaneously grows while the price of a good goes up, he may still buy more of that good, despite the price increase. Thus, those 6% are not instances of the failure of the law of supply and demand, but are examples where an additional factor was at work.

It is impossible to go over all the numbers, since the population is infinite in this case (series of all possible (F,m) ), so how can we ever know that this is true by using samples of it?

Note: There is a difference between the concept of probability of population and actually having correct knowledge about the population. In statistics we seek knowledge of true probability, while in my F=ma example I was asking about true knowledge of the population, and not probability of it.

Ifatart is correct. Using statistics alone, one can never be certain of one's conclusions. The problem with treating concepts such as the laws of motion or the law of supply and demand as knowledge to be validated by statistics is that such a standard invites skepticism. If we are only relying on statistical correlation, at what point can you say that you have certainty? No such point exists. In statistics, we can only say that with a large enough sample size that we are 99.9999% certain that a valid correlation exists. Such a method cannot prove anything. To establish proof, statistics can be a valuable tool (as explained above), but proof will only come through conceptual understanding of the cause and effect mechanism, validated by sufficient examples (which need only be enough to validate the concept -- e.g.: the billiard ball example).

Proof does not necessarily require a "statistically significant" number of examples. Conversely, a statistically significant sample size does not constitute proof.

When one gains certainty through a proper understanding of the cause and effect mechanism, one doesn't have to be scared off by seeming contradictions brought up by statistics. For example, using the supply and demand example, one doesn't need to be too worried about the 6% "failure" rate of supply and demand. One knows supply and demand is a valid principle through the conceptual understanding already established, so in the face of that 6% one suspects there must be another factor at work in those instances. Sure enough, you discover that there is, the "income effect".

To use the billiard ball example, one can be certain that if a billiard ball is hit a certain way, the ball will move in a particular direction. If that doesn't happen, one can suspect that there is another factor at work, e.g.: a rigged ball, magnets, a tilted table, a pool cue with a spongy tip, etc. When you look around in the smoke-filled pool hall and see people pointing at you and laughing at your loss of the $1,000 pool bet, you should suspect it might not be an honest table. And don't rely on statistics (i.e., achieving a credible "statistically significant" sample) to reach that conclusion. If you do, you will become poor very fast!

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The virtue of statistics is that it provides a precise measure of the degree of correlation. It gives precision (a measure of the degree of correlation) to something that remains unknown (the exact causal relationship among phenomonena).
I agree, with the essential caveat (one which is forgotten too often) that the mathematical computation can only address the correlations that you ask about, and cannot provide the correlations that you failed to ask about.
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As you know, an inference from a sample to properties of the population is mathematically valid only if all members of the population have an equal chance of being included in the sample.

It is never about conducting enough experiments: it is always about dispelling reason to doubt.

...

Whenever you extrapolate from obversation to non-analogous circumstances, you are conjecturing, and untested conjectures are a sign that your law may not be valid, insofar as it is based on an unvalidated assumption (which might be something like "God designed the universe to be mathematically elegant").

Thank you for a great, informative reply.

Just one question now: is "combinatorical probability" that you talked about in earlier posts the same as "true probability" (of the population)?

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