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The Nature of Probability

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After searching the forums for the term "probability" I didn't come up with anything directly addressing this question. If you know of an existing thread that does, I would appreciate being pointed there.

I am currently taking a "Foundations of Probability and Inductive Logic" philosophy class. I am trying to figure out what the NATURE of probability is from an Objectivist standpoint. What does it refer to? Is it a measure of a metaphysical relationship of possible outcomes or is it somehow a measure of epistemic relations? If so, what are the means of inducing these relationships?

I very much enjoyed Peikoff's recent podcast where he tells the story of trying to figure out how many people would be in the lobby of Ayn Rand's apartment building and it started to shed some light on this question but I am looking for a more explicit and succint response to the questions I listed.

When being faced in my class with interpretations of probability such as Richard von Mises Frequency interpretation, Karl Popper's Propensity interpretation and F.P. Ramsey's Subjective interpretation, what is the proper way of viewing this subject?

Thanks for any help you guys can offer.

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After searching the forums for the term "probability" I didn't come up with anything directly addressing this question. If you know of an existing thread that does, I would appreciate being pointed there.

I am currently taking a "Foundations of Probability and Inductive Logic" philosophy class. I am trying to figure out what the NATURE of probability is from an Objectivist standpoint. What does it refer to? Is it a measure of a metaphysical relationship of possible outcomes or is it somehow a measure of epistemic relations? If so, what are the means of inducing these relationships?

I very much enjoyed Peikoff's recent podcast where he tells the story of trying to figure out how many people would be in the lobby of Ayn Rand's apartment building and it started to shed some light on this question but I am looking for a more explicit and succint response to the questions I listed.

When being faced in my class with interpretations of probability such as Richard von Mises Frequency interpretation, Karl Popper's Propensity interpretation and F.P. Ramsey's Subjective interpretation, what is the proper way of viewing this subject?

Thanks for any help you guys can offer.

I don't know what you have read but Peikoff discusses in OPAR how knowledge progresses from the possible to the probable to certainty in a context, based on the quality/quantity of evidence gathered.

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Thank you for the response. I have already read OPAR and that was the first place I went when looking for an answer to this. It wasn't any help with regard to what my question is. Specifically I am looking for how probability should be quantifiably measured (if it even can) and what the context of its use would be. The aspect you mentions only helps minimally with the basic concept of probability but doesn't really address the nature and mathematics.

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Perhaps it would also be helpful if I gave an example. This is the question we have to write our final term paper on and address each of the different interpretations I listed in my OP and how they would answer it. So if someone could shed some light on how the Objectivist interpretation of probability would answer this (or maybe it would completely reject the question), I think that would be very helpful to my understanding:

You have a biased coin (i.e. it's either weighted towards the heads side or weighted towards the tails side). But you don't know which side it is baised towards. What is the probability of a flip coming up heads?

Edited by KevinDW78
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You have a biased coin (i.e. it's either weighted towards the heads side or weighted towards the tails side). But you don't know which side it is baised towards. What is the probability of a flip coming up heads?

How do you know it is biased? Divine revelation? Did someone hand the coin to you and say, "this coin is biased, but I'm not saying which side it's biased toward." How do you know they're telling you the truth? How do you know they're not delusional themselves?

Basically, *knowing* the coin is biased means knowing *how* it is biased - ie, knowing which side it is biased toward. And the only way to know that is through test flips.

Probability is an attempt to predict events in the context of limited knowledge. For example, if you know a meteor is headed in our direction, the probability of it striking our planet depends on the accuracy of our knowledge of the meteor's mass, position, and velocity, among other things, in order to model its trajectory through the solar system. The probability likewise also depends on the accuracy of the model.

It's easy for people to get confused and think that the probability depends on things in reality - such as the actual size/velocity/mass of the meteor - but if we knew all those numbers exactly, then we could definitely say the probability is either 0 or 1. We can't know those numbers exactly, and certainly can't model the solar system exactly - so really the probability depends on our knowledge, on our measurements.

Likewise, probability is meaningless to an omniscient being, who could only say, "it's going to hit the Earth" or "it's not going to hit the Earth."

Getting back to the coin example, you really have no way to model coin flips, so the best you can do is to just do a bunch of flips and see what the fraction of heads/tails is. You can't do the same thing with meteors starting from the same initial position/velocity/mass as the observed meteor, so you have to work with what's possible - making measurements and doing models.

Edited by brian0918
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You have a biased coin (i.e. it's either weighted towards the heads side or weighted towards the tails side). But you don't know which side it is baised towards. What is the probability of a flip coming up heads?

Here's my question:

Does 'subjective' mean that you would have to say 1/2 probability of heads, because you (the subject?) don't know which side is weighted? Because that seems like a damn objective way of assigning the probabilities, based on one's context of knowledge, to me.

And my answer:

Since all knowledge is contextual, in the context of not knowing which side is weighted, the answer is 1/2. If your context changes, the (objective!) answer changes.

Edited by Jake_Ellison
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You have a biased coin (i.e. it's either weighted towards the heads side or weighted towards the tails side). But you don't know which side it is baised towards. What is the probability of a flip coming up heads?

I don't know whether that would even be a factor in a coin toss. For example, if you set it at the edge of a regular table with tails facing up, the odds are nearly 1.0 it will show heads upon hiting the floor. That's because in the time it will take to fall that short distance it will only make half a revolution (this is the infamous buttered toast problem, BTW, which explains why almost always the buttered side hits the floor).

Anyway, assuming such bias has any physical effect, then you can't possibly estimate the odds for heads unless you know which way it's biased.

As for probability, it is a mathematical measure of the likelyhood a given event will take palce. As Brian said, the measurement depends on the accuracy of your knowledge. In card and dice games the odds can be calcualted with complete precission, since all factors are known (how many cards of each type there are, how many are dealt and when, how many sides for each die, etc). In fact, gambling is a breeding ground for statisticians (look up The Wizzard of Odds in the web if you want mroe info).

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The word "probability" refers to three concepts, and they are related concepts -- one might say that they are "senses" of a single concept. This is a complicated issue in the theory of concepts, which you don't have to fully engage for this paper, but you're nicking the corners of the problem.

The three concepts pertain to combinatorics, frequency of occurrence, and evidence. There are 6 logically possible outcomes of rolling a standard die, and 36 outcomes if you roll it twice -- the combinatoric sense. The frequency sense is empirical, and is seen in statements like "the probability of an American owning an SUV is 1 in 3".

The evidentiary concept, which is most salient for Objectivism, relies on the other two senses, and amounts to the strength of your knowledge of causal factors for an event. If you know all of those factors, then you can say in advance of the event what the outcome will be. Therefore if you know that a coin is heavily enough weighted on the heads side and you know enough about coin-tossing physics, you can predict that the coin will always land tails-up. You can compare that prediction to observation of events and see that your prediction is correct.

You can also use combinatorics to inform you that observational frequency does not match logical possibilities (i.e. that there are two metaphysically possible outcomes), which is evidence for an alternative conclusion (there is only one metaphysically possible conclusion). This mismatch is thus evidence.

If I am handed a coin, I have no initial reason to believe any particular conclusion about whether it is weighted on one side. Any conclusions at this point are arbitrary. By tossing the coin once and observing that it came up tails, I now have to integrate that observation with everything that I know, and I have evidence against a conclusion that the coin is strongly weighted in favor of heads. My evidence is consistent with the conclusions "is not weighted" and "is weighted in favor of tails". After a second toss coming up tails, the evidence against "strongly weighted in favor of heads" is strengthened -- we have moved that (negative) conclusion further on the evidentiary scale in the direction of "proven", i.e. "certain". At some point with, say, 5 heads in a row, I now realize the magnitude of the mismatch between the combinatorics of the "neutral coin" theory, and point to these observations as evidence proving the conclusion that the coin is weighted in favor of tails.

With perfect knowledge of causal factors, there will be no variance in events -- you will always observe the particular outcome in the presence of those causal factors. When there is variance, that means that you do not know the causal factors perfectly. That means that when I find variance, I have evidence that I do not know the causal factors perfectly, and if I can find a correlation with some new factor, then I may have evidence for adding some new causal factor to my model of the world.

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In card and dice games the odds can be calcualted with complete precission, since all factors are known (how many cards of each type there are, how many are dealt and when, how many sides for each die, etc). In fact, gambling is a breeding ground for statisticians (look up The Wizzard of Odds in the web if you want mroe info).

If all factors were known exactly, you would be able to say that the next card turned up would definitely be a king of hearts, but shuffling is difficult to model accurately, so that potential factor is ignored. If you knew that a dealer tended to shuffle in some way that is predictable, you could add that knowledge to get better odds than simple deck probability dictates.

Edited by brian0918
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As for probability, it is a mathematical measure of the likelyhood a given event will take palce. As Brian said, the measurement depends on the accuracy of your knowledge.
What is "the likelihood of an event"? If an event takes place, is it highly likely and if it does not take place is it highly unlikely? If you do not know whether an event took place, does that affect the likelihood of the event?
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If all factors were known exactly, you would be able to say that the next card turned up would definitely be a king of hearts, but shuffling is difficult to model accurately, so that potential factor is ignored. If you knew that a dealer tended to shuffle in some way that is predictable, you could add that knowledge to get better odds than simple deck probability dictates.

As far as I know the various models assume shuffling is random, meaning initial card order is unkown, and therefore all cards have the same odds of appearing first. Granted different people shuffle differently and may bias the deck somehow, you'd still have to deal with the initial shuffling conditions (from a pristine deck, in which the cards are in a known order, or from a used deck, in which cards are not in any particular order), and the fact that often one of the players cuts the deck.

This reminds me of biased wheels. An ideal wheel is balanced and perfectly horizontal. Over time it may unbalance and tilt slightly to one side. You could then measure the frequencies of each number and conclude whether any numbers come up more frequently than they should, all without measuring the tilt or balance of the wheel. So maybe you could simply observe a dealer and determine a probable card order over time.

In dice games there are such things as loaded dice and biased dice. If you own a backgammon set or a Monopoly game, you'll notice after years of use the dice tend to come up with some combinations more frequently than random chance would explain. To combat this casinos change dice frequently, ensuring as random a throw as it can possibly be. Then there's the very controversial matter of dice control.

The thing about cards is that the dealing of cards is a sereis of related events, whereas in wheels or dice there's only independent events. Once the ace of spades is dealt, it cannot appear again (in a one deck game). Card counting works best one on one against a dealer in a one deck game (try and find one!)

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What is "the likelihood of an event"? If an event takes place, is it highly likely and if it does not take place is it highly unlikely? If you do not know whether an event took place, does that affect the likelihood of the event?

Sorry, that whoudl be the likelyhood that an outcome will take place.

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Sorry, that whoudl be the likelyhood that an outcome will take place.
Okay, what is "the likelihood that an outcome will take place"? If there is a certain outcome, is that outcome highly likely and if it there is a different outcome is it highly unlikely? If you do not know the outcome, does that affect the likelihood of an outcome? I'm trying to understand what kind of metaphysical claim you are making.
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Thanks for the replies. This is definitely helping. From what you guys are saying though, it would be in line with Richard von Mises Frequency interpretation. i.e. you can't say anything about the probability until you start flipping and accumulate data. I would have initially agreed but then I learned that Richard von Mises is one of the people who founded the theory of Logical-Positivism, which Rand fervently and explicitly rejected. So Then I started to get confused and torn as to where my thoughts on this should go.

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From what you guys are saying though, it would be in line with Richard von Mises Frequency interpretation. i.e. you can't say anything about the probability until you start flipping and accumulate data.

Assuming it's impossible for you to model or understand anything, then yes. For the coin flip example, without knowing anything about the composition of the coin, you have to assume the initial probability of 0.5 heads, 0.5 tails. You could examine the coin in detail, run mass spectrometry on it and determine that one half is lead, the other half is aluminum, then you would know more about it and be able to adjust your probability. But if you're entirely limited in your examination and ability to model, you have to assume 50/50 for the probability. For coin flips, it's easy to gather information by doing test flips, so there's not really a reason to model. For meteors, it's not possible to do similar "test flips", so all you have are models.

Let's say you have a coin about which you know nothing, but after flipping 10,000 times, you get 70/30. You can almost certainly say that you're observing a metaphysical truth (e.g. observing a stick bend when it goes in the water), but in order to understand why that is true - ie, why the probability is not 50/50 - you have to examine the coin in more detail, you have to learn more about reality.

I would have initially agreed but then I learned that Richard von Mises is one of the people who founded the theory of Logical-Positivism, which Rand fervently and explicitly rejected.

You're giving him too much credit for consistency if you believe that he's only capable of forming true or false conclusions.

Edited by brian0918
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But if you're entirely limited in your examination and ability to model, you have to assume 50/50 for the probability.

From what I gather than, Objectivism would be more in line with Karl Popper's Propensity theory since Propensity theory is Frequency theory but with the addition that you can incorporate priot knowledge of the universe before you start flipping. von Mises would have rejected even that you can assume 50/50. He would flat out reject the scenario and say it is outside the purvue of probability until only AFTER you started flipping. But Popper would have said (if I understand his theory correctly) that we can at least assume it is 50/50 because we have prior knowledge about coins and what happens when we flip them, so we can start with an assumption based on that prior knowledge, than incorporate more data (via Bayes' Rule) as it becomes available to us to adjust the calculation.

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There have been a lot of interpretations here of what Probability means and how it is to be applied.

But if the original question was, in part, asking for the Objectivist position on Probability, I would say that that is not a philosophical concept. It is simply a measure of some level of evidence for something. It is, in essence, measured against "certainty," a more important concept.

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From what you guys are saying though, it would be in line with Richard von Mises Frequency interpretation. i.e. you can't say anything about the probability until you start flipping and accumulate data.
I don't know what R. von Mises specifically claimed, but given that probability is about degree of evidence for something, and "flipping and accumulating data" just stands for "making observations of something", then this must be true, since proof of a conclusion requires sensory data.

As Brian said, that need not be limited to coin tossing -- there are many other ways to prove (or disprove) the conclusion that there's something funny going on. You will have to actually inspect the physical structure of the coin to determine that it's a property of the coin, and if so, whether it's about weight. Without that, the only conclusion that you can reach with certainty is that something is screwy. You cannot even reach that conclusion without prior knowledge (since the number of states that the coin can be in is not given a priori) -- you must incorporate prior knowledge of the universe. (The problem for Popper is that you can't be certain of any of that knowledge).

However, as I understand Popper's theory, it equates probability with physical propensity. That's at odds with the Objectivist view, which is that probability is about proof, thus is epistemological.

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Okay, what is "the likelihood that an outcome will take place"?

The odds that an outcome will take place. For example, if you throw a standard six-sided die with the faces numbered one through six, each number has odds of 1/6th of appearing. If you throw two such dies, the odds of the same number showing up on both is 1/36th, but the odds of throwing any two numbers adding up to seven is 6/36th, or 1/6th. Therefore seven is a more likely throw than a double number.

In the case of a coin toss, the odds are 1/2 for each side, mening either outcome is as likely as the other.

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Statistics can lead to a hypothesis but an inference of a mechanism of causation is necessary for a valid induction. This comes from comments made by Dr. Peikoff in two sources.

see Notes on The Art of Thinking Lecture 6 Certainty

Three aspects of certainty covered are predictions about the future, statistics, context and error. This is also the original appearance of the Rand story about statistically modelling the number of people in the lobby of the building. An additional distinction made is "metaphysical possibility" versus "epistemological possibility". In question periods on "Art of Thinking" he concedes statistics as used in insurance companies can be useful but that they lead only to pre-scientific guesses or hypothesis, not actual inductions.

From the first lecture of "Notes on Induction in Physics and Philosophy" Dr. Peikoff states that statistical measurements are not a basis for induction.

Generalization - The inference of some member of a class to all. The inference moves from

* from the observed to the unobserved

* from the past to the future

* from here to everywhere

Generalizing is the essence of human cognition and distinguishing feature of man from the animal.

Induction is the primary process of gaining knowledge that goes beyond perception (generalizing). Deduction also moves beyond perception but presupposes premises; therefore is not primary.

Definition of Generalization - a proposition that ascribes a characteristic to every member of an unlimited class, however a member is placed in time and space. Format: "All S is P"

'unlimited' is in the definition to rule out simple inventory as an induction. Example: Inspecting every marble in a bag of marbles, seeing they are all red, then stating "All the marbles in this bag are red" is not an induction despite the universal format. {this is not an open-ended universal because "all the marbles in this bag" is not a concept it is a concrete}

Generalizations are made possible by man's conceptual faculty. S and P are concepts. {Definition of concept from the Lexicon}

Concepts are tools of knowledge, file-folders. They are not the claims to knowledge, they organize it and integrate it. Higher level concepts can presuppose knowledge but themselves state nothing.

'Table' is not true or false but valid or invalid.

'All S is P' is true or false and belongs in the S file-folder

Method of enumeration is false. Enumeration is merely counting instances, and supposedly the bigger the count the greater the probability. No certainty is possible. Subject to problem of the swan's. Enumeration provides no basis of moving from some to all. A true generalization can be formed from a single observation, there are false generalizations based on millions of instances.

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Completely 'skewed' results, with an UNbiased coin, are not unusual. It might take 100s of thousands of flips to achieve the 50/50 proposition.

The Law of averages can take a long time to prove itself. Which is why a sampling of just 5 flips (as someone suggested) is far too small to establish anything.

There have been many gamblers who went broke, patiently waiting for the roulette wheel to come up with (say) 12 reds in a row, and betting big on black - under the fallacy that the odds are hugely in favour of Black coming up on the next spin !!

The probabilities of each toss, or spin, remain the same, ie. 1:1 , REGARDLESS of the previous result.

As they say, dice - or cards - have no memory. (A good case for objectivity, vs. intrincism, in gaming.

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It is simply a measure of some level of evidence for something. It is, in essence, measured against "certainty," a more important concept.

But that just takes me back to square one. HOW does one measure evidence against certainty? By what standard? This is why this question has been so difficult for me to grapple with.

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The odds that an outcome will take place. For example, if you throw a standard six-sided die with the faces numbered one through six, each number has odds of 1/6th of appearing.

That's not true. It is a fact that the die will land on a specific side, as long as you throw it in a specific way. After it was thrown, if there is no changing the side it will land on is settled, and even knowable. What is causing people in casinos to not know which side that is, is their inability to study the die, and the act of throwing it, between the very inexact throw executed by the human hand and the landing of the die.

If I had that ability, I could tell you what side the die is going to land on, with certainty. It can only land on the side it was going to land on, there is no chance of it "changing its mind", and landing on another side. There are no odds.

What is true, is that if thown repeatedly by a person, a balanced die will land on each side about 1/6th of the time. Using combinatorics, you can then improve your chances of guessing what number two dice will give you, in the absence of any knowledge of how the dice were thrown. In other words, seven would be a better guess than two, unless you are able to actually figure out which number it is going to be, with certainty or at least with greater precision (realistically, that would be easier to do if the throw was executed by a device, not a hand).

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The odds that an outcome will take place.
What are "the odds that an outcome will take place". I'm hoping that at some point, you will define "probability" or "likelihood" or "odds" in terms of something tangible or perceptible, or in some other way explain your view of the metaphysical nature of "probability".

For example, you claim that the "odds" for a particular coin toss is 1/2, without justifying that claim by referring to something in reality.

Anyhow, in fact "probability" is not just formal combinatorics.

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