It seems like in everyday probability (not quantum physics), probabilities are really just a substitute for an unknown. Take a coin flip for example. We say it's "random," a 50% change of head and a 50% chance of tails. However, if I knew exactly the density, size, and shape of the coin; the air density; with how much force the coin flipped; where exactly that force was placed; the distance of the coin to the floor; etc., wouldn't I be able to predict, using basic physics, with 100% accuracy whether it would land on heads or tails? If so, isn't probability in this scenario just a way for me handle incomplete information?

Isn't it the same thing if I shuffle a deck of cards (which is what got me thinking about it)? I treat the order of cards as random because I don't know what the order is, but it's not as if there really is a 1/52 chance that the first card I draw is the Ace of Spades--it either 100% is the ace of spades or 100% isn't.

If rolling a die and shuffling a deck isn't really random, wouldn't it follow that computerized random number generators aren't random either, since if I know the algorithm (and probably a few other variables) I'd know what the number is going to be?

Thanks in advance to anyone who takes the time to answer, especially a noob question from a non-math person like myself. I didn't want to go on reddit because a lot of those people pretend to be knowledgeable but aren't. Some additional meta-remarks:

First, I know there is a similar question already been answered Random vs Unknown. So please, don't refer me to that. I think the question I'm about to ask is much more narrow and grounded in much simpler math.

Second, I'm not a math person, so please stick to simple examples and non-technical language (unless absolutely necessary, in which case pretend like you're explaining yourself to a moderately intelligent senior in college majoring in art history).

Third, I have a good understanding of ELEMENTARY probability. This is mostly because I play a lot of poker, but I understand how odds in other gambling games work like roulette, dice, lotteries, etc. Again, this is very BASIC stuff so please no quantum physics if it can be avoided.

Fourth, not to sound callous, but I want people to discuss the answer to my question and not show me how much more they know then me. I say this because I've seen people try "beat" someone in an argument by purposefully using needlessly hyper-technical language and confusing the other person with their vocabulary rather than debating the actual question. For example, instead of saying "it would behoove you to ingest some acetylsalicylic acid" say "you should take some aspirin."

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    $\begingroup$ There are several different schools of thought on interpretations of classical probability (and, of course, contention) and much interesting literature on these. en.m.wikipedia.org/wiki/Probability_interpretations is a good start. Same goes for quantum probability. $\endgroup$ Jun 23 '17 at 2:56
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    $\begingroup$ See some related discussion in the philosophy Q&A: philosophy.stackexchange.com/questions/29364/… . It's possible "true" randomness only exists at the quantum level, and for everything above that, events are only random given the information we have (or don't have) available. Your wording "It seems like in everyday probability (not quantum physics), probabilities are really just a substitute for an unknown" seems like a good way to express that idea. $\endgroup$
    – Marius
    Jun 23 '17 at 4:18
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    $\begingroup$ More than 50% your question text are meta-remarks that are not helping to formulate the question. They were preceding the actual question, which made the post a bit hard to digest. I took the liberty to move them all the way down, after the actual question. To be honest, I think this whole section can be erased, but that's up to you. +1 for the question itself. $\endgroup$
    – amoeba
    Jun 23 '17 at 8:13
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    $\begingroup$ @Marius +1 for the link and for the summary. I'd only add that the nature of randomness on the quantum level is debated as well. $\endgroup$
    – amoeba
    Jun 23 '17 at 8:15
  • $\begingroup$ amoeba, I appreciate you moving the section down, but I wouldn't want it erased. I felt the first point was necessary because I really think somebody would've just linked me to that question. The second and third were necessary so people would understand I have virtually no knowledge of math beyond the basic concepts and adjust the explanations accordingly. The fourth is the least necessary, but I do think it prevented some answers from using terms I'm not familiar with. $\endgroup$
    – N00ber
    Jun 23 '17 at 21:43

You are perfectly right, probability is the measure of uncertainty. Coin flip is a nice example, as discussed in another thread. Tossing a coin is a physical, deterministic process. In fact there are people who have learned to flip the coin in such way to get the outcome they want and are machines that produce deterministic, predictable coin flips. Let me, once again, quote E. Borel (after Bruno de Finetti, Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science):

"One can bet, in heads or tails, after the coin, already tossed, is in the air, so that its movement is determined. One can also bet after the coin has landed, on the sole condition that one does not see on what side it has landed. Probability does not lie in the fact that the event is undetermined (in the more or less philosophical sense of the term) but only in our inability to predict what possibility will take place, or to know what possibility has taken place."

To make things even more complicated, there are Bayesians who interpret probability as degree of belief. In fact, there are many different interpretations of probability. When something is impossible, or very, very unlikely we assign zero probability to it (check here, here and here), when it is certain, the probability is equal to unity. When talking only about impossible and unlikely events, probability reduces to logic. When considering uncertain events, it may be seen as an extension of logic.

But probability is not a substitute for "unknown", it is a measure of how much "likely" the unknown is. It may be interpreted in different ways, and so measure slightly different things, but in the end it lets us to quantify the unknown. Probability lets us say much more about the reality, then that something is "unknown", or "uncertain". But it is not only about measuring, probability lets us to make predictions, precisely estimate the expectations and risks, or apply Bayes theorem to combine probabilities, to give only few examples. In fact, as shown by Daniel Kahneman and Amos Tversky, people are poor in reasoning about uncertainties and risks, while using formal, probabilistic reasoning guards us from our biases.

  • $\begingroup$ +1. Very nice, and with a lot of links to (good) further reading. $\endgroup$
    – amoeba
    Jun 23 '17 at 11:39
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    $\begingroup$ Definitely would give this a +1, except for "To make things even worse, there are Bayesians . . ." $\endgroup$
    – Darren
    Jun 23 '17 at 13:42
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    $\begingroup$ @Darren "to make things worse" is irony, if you look at the linked threads you'll notice that there are several answers of mine that discuss the Bayesian approach. I'd consider myself as a Bayesian by heart. $\endgroup$
    – Tim
    Jun 23 '17 at 13:48

There is a long and deep history of uncertainty and the quantification of uncertainty, with terms like "subjective probability." A key result is Cox's Theorem. He posited three properties of any measure or representation of uncertainty:

  • Divisibility and comparability – The plausibility of a proposition is a real number and is dependent on information we have related to the proposition.
  • Common sense – Plausibilities should vary sensibly with the assessment of plausibilities in the model.
  • Consistency – If the plausibility of a proposition can be derived in many ways, all the results must be equal.

These make perfect sense, and capture what we want in any representation of uncertainty. Together with derived results, such as the probability of a proposition $A$ plus the probability of the proposition not $A$ must sum to 1.0 (certainty), and that uncertainty is monotonic (if you have more and more uncertainty, the numerical value describing that certainty must only get smaller), he derived mathematically what any such representation must obey. His result is that they must be expressed in, and use the relations of probability.

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    $\begingroup$ I think I understand the propositions: (1) whether any proposition, P, is true is a number from 0.0 to 1.0, (2) you should use common sense (i.e. basic logic) when you assess the likelihood of P within any given system, and (3) if there a many ways to get a result, all the results must be the same. However, I don't see how this answers my questions. Also, what's the difference between plausibility and probability? $\endgroup$
    – N00ber
    Jun 23 '17 at 1:33
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    $\begingroup$ This just seems to describe how a system of probabilities should function, but I'm asking about the thing probabilities represent. $\endgroup$
    – N00ber
    Jun 23 '17 at 1:41
  • $\begingroup$ Cox's results is that every form of uncertainty—plausibility, subjective probability, confidence, etc.—is ultimately expressible in the language of probability, and as such are fundamentally unified. We have lots of variations in our terminology within natural language (including between different natural languages) but when you ultimately wish to calculate something and do an experiment, you must use terminology of probability. What his results show, too, are that the concepts of "fuzzy logic" (when they differ from probability) do not advance our understanding of uncertainty. $\endgroup$ Jun 25 '17 at 6:41
  • $\begingroup$ I just read your response again, and it actually does answer my question, albeit in a way that is unnecessarily difficult to understand. $\endgroup$
    – N00ber
    Jun 29 '17 at 4:40

The short answer is yes. The first chapter of this phd thesis has an example with a simulation of flipping a throwing pin. The outcome 'pin-up' or 'pin-down' depends on a number of variables (like rotation speed and size), which we do not usually control in everyday life. So in the simulation the system is deterministic: given the input variables the outcome can be computed. But when flipping a pin on your table, you don't know the exact values so you can only estimate the probability of the pin landing 'pin-up' or 'pin-down'.

As a final remark we simply note that most, if not all real-world systems can be described (at least in principle) in terms of a dynamical system, and that our interpretation of ‘random’ as arising from uncertain, incomplete knowledge about the state of a system applies even down to the quantum level.


Talking quantum physics might nevertheless help to appreciate certain issues and paradoxes. Take for example lemur’s comment:

..., but these hurt my philosophical feelings: QM is Nature’s way of having to avoid dealing with an infinite number of bits

But there is a paradox here, since it seems that Nature still requires an infinite number of bits, just to write down the exact probability of an event. The same issue happens for everyday probabilities: The weather forecast may predict the probability of precipitation for the following day in a certain area during a certain time span to be 30%. But how accurate is this probability? Does it mean that the actual probability is between 25% and 35%? Does it even make sense to talk about the accuracy of a probability? The probability for a certain number in Roulette is 1/37, but can one also say something about the accuracy of that probability? Here one can at least test the hypothesis about a given accuracy of the probability by performing a sufficient number of repeated experiments.

Even if not meant that way, Pascal's Wager presents a similar type of paradox. It describes an experiment which can not be repeated, and then assumes that one could assign a probability like 0.000001 or 1e-3000 to a certain outcome, without questioning whether such an accurate probability even makes sense in this context.

A paper by Ole Peters and Murray Gell-Mann (the famous physicists) triggered those thoughts...

  • $\begingroup$ Probability per se cannot be "accurate" or not, I guess you mean some kind of estimates of the probabilities..? You can talk about accuracy of forecasts, or accuracy for uniform model of roulette outcomes etc., but this is not accuracy of probabilities. $\endgroup$
    – Tim
    Jun 23 '17 at 22:00
  • $\begingroup$ @Tim I mean the concrete situations I list where it is common to state some probability. In QM one can compute probabilities for certain outcomes, the weather forecast states some probability of precipitation, there are probabilities in Roulette, and Pascal's Wager assumes that there is a probability that god exists... I do think that some situations allow more accurate probabilities than others (mainly based on how often and how faithful experiments for testing the probabilities can be performed and repeated). $\endgroup$ Jun 23 '17 at 22:25
  • $\begingroup$ But you are talking about estimated probabilities. $\endgroup$
    – Tim
    Jun 23 '17 at 22:30
  • $\begingroup$ @Tim I am thinking more about testing probabilities (for a given accuracy), then about estimating probabilities. The testing relies on additional properties like independence, but hopefully not on identically repeated experiments (otherwise the probability of precipitation could never be tested, for example). I come from a logical background, and have something similar to the game semantics from predicate logic in mind. But my answer here really just consists of the listed situations, not of what I have in mind or think myself about possible resolution of these paradoxes. $\endgroup$ Jun 23 '17 at 22:49
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    $\begingroup$ But those properties, that you're talking about are properties of statistical models, not probabilities. Example: imagine a fair coin with probability heads = tails = 0.5. The probability in here is 0.5. There is no accuracy that can be measured in here. You can toss it a number of times and compare the estimated probabilities given the data with value 0.5, but this will tell you only about precision of measurement and your estimates. $\endgroup$
    – Tim
    Jun 24 '17 at 7:35

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