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Probabilities are usually considered to be a real number between 0 and 1. A real number has an infinite decimal expansion. Are probabilities really real numbers? Is the infinite decimal expansion really physical? Ordinarily, when we deal with probabilities in practice, we only deal with the first few significant digits. Are the later digits physical?

If one wants to measure the 100th decimal digit of a probability, one has to sample an ensemble of size slightly over $10^{200}$ to measure the right answer with probability near 1. Does the necessity for such a large ensemble mean thinking of probability as a real tangible physical quantity is incorrect? There are two cases to consider here. If the probability is "actually" around say $4.83 \times 10^{-100}$, then one would expect the 100th decimal digit to be a lot more physical compared to the case where the "actual" probability is say $.804\cdots 3\cdots $, if such a precise probability even makes any sense. But even in the former case, isn't the probability effectively zero for all practical purposes? Effectively, isn't $3\times 10^{-145}$ indistinguishable from $7\times 10^{-82}$? How can one distinguish between both cases in practice?

If probabilities are even more tangible than that, shouldn't we be able to set up a gadget which behaves in one way if the 100th digit is even, but in a different manner if it's odd? That's just not how probabilities behave in practice.

This question leads up to the nature of the complex coefficients of the wave function in quantum mechanics. Are they really physical tangible complex numbers? What about their absolute square (a real number), or relative phases? What if we set up a case where we have a nearly exact destructive interference, with the coefficients of two basis terms nearly cancelling up to $10^{-50}$?

In a Bayesian sense, it is ridiculous to suppose our knowledge or ignorance of a system can be quantified so exactly. If considered as betting strategies, a typical "rational" agent might as well flip a coin to decide between bets when their expectation values match up to the 100th digit. In a frequentist sense, an ensemble of size at least $10^{200}$ is needed. Only propensitist interpretations can make the real value of a probability physical.

Theoretically, for idealized systems, one can have actual real numbers with an infinite decimal expansion for probabilities, but do such concepts apply to the real world? If not, then what are probabilities really?

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Probability falling between 0 and 1 is an artifact of the way problems are conceived. It represents something that can become known as a process moves forward in time. Example: a binary decision point in a program having two branches. If each branch has P = 1/2, then after the point is traversed, the state contains 1 bit more of information. Conversely, if the two branches join together again, 1 bit of information is lost, corresponding to a probability of 2. –  Mike Dunlavey Sep 21 '12 at 13:58
    
@MikeDunlavey: Your statement is with probability 2 meaningless. –  Arnold Neumaier Sep 21 '12 at 15:41
    
@Arnold: OK. Define $P(B|A)=P(AB)/P(A)$. It is >1 if events can merge (or run backward in time), not simply branch forward. Happens all the time in algorithms. They gain information, then lose it. If $I = -log(P)$, and $I < 0$, then $P > 1$. Not a deep idea, actually. –  Mike Dunlavey Sep 21 '12 at 19:04
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@MikeDunlavey: P(AB) is the probability that $A$ and $B$ happen (in a particular situation), and is almost at most $P(A)$, the probability that $A$ happens (in precisely the same situation). Thus $P(B|A)\le 1$. If you mix up the situations (e.g., by considering the numerator and denominator at different times), you get nonsense not covered by the mathematics. –  Arnold Neumaier Sep 22 '12 at 11:39

4 Answers 4

All concepts one works with in theoretical physcis are idealizations. One needs idealizations in order to treat a system in a mathematically adequate way.

In particular, probabilities, are like any measurement values idealized objects that can take arbitrary real numbers (in [0.1] for probabilities) as values. Restricting them to be rational (as relative frequenices are) is arbitrary, and awkward in theoretical arguments.

Probabilities are thought to be limits of relative frequencies in case one would be able to repeat events infinitely often, in the same sense as the side or diagonal of a physical square are thought to be limits of measurements with higher and higher precision. In statistics, independent and frequent repetition increases accuracy, to $O(N^{-1/2})$ accuracy with $N$ independent repetitions.

Very tiny probability are nothing exceptional; indeed, they are the norm for complex events. Most complex events that happen have an extremely tiny probability.

For example, if you record the results of throwing a die 1000 times in a row (an easily performed experiment), the probability that you obtained the sequence you actually recorded is only $6^{-1000}\approx10^{-778}$, and we can know this probability to as many places we like to consider. This is far smaller than the probability for picking a random atom form all the atoms in the vcisible universe - but the event actually happened!

The sequence just considered is however not repeatable - to get the same sequence again is very unlikely.

This is why in science, which mainly studies repeatable events only, tiny probabilities can usually be neglected (unless very many of them must be summed up).

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First of all, it's good you're talking about $10^{200}$, that you squared the inverse probability precision. It's because the relative error of a number measured statistically scales like $1/\sqrt{N}$ where $N$ is the number of repetitions.

Otherwise, it's not clear why you singled out probabilities. Any quantity in physics, e.g. the radius or mass of a star or an atom, may only be measured with some error margin. Certain values of any quantity that are too close to zero are indistinguishable from zero in practice. For example, the radius of the electron, its "inner structure", seems to be zero experimentally. But we know it may still be a string of size $10^{-35}$ meters and even much larger than that.

Experiments – and all considerations "in practice" – are always limited by a finite precision. However, theories may still predict the values, whether they're radii or probabilities, with a better precision than what we can measure at a given moment. Of course, there may always be doubts about the validity and precision of each theory until we're actually able to verify its predictions. But we could also get certain about the validity and accuracy of a theory even if a particular prediction of it couldn't be measured.

Probabilities such as $10^{-145}$ and $10^{-82}$ are very small and indistinguishable from zero if they quantify the probability of an overall, special event (without too many "analogous" events) that may happen but doesn't have to happen throughout the whole "history". Indeed, if the probability of a collision of two large planets (before the Sun goes red giant) were $10^{-145}$ or $10^{-82}$, we could approximate both of them by zero. We could say it won't happen.

In such a case, it would be very hard to measure such a probability in the frequentist fashion because we would have to repeat the same experiment in $10^{290}$ or $10^{164}$ Solar Systems. There aren't this many stars. Most likely, the total number of star births in the history of our visible Universe will never be this high.

However, if we are talking about the probability of a completely arbitrary event, one that may be repeated many times, $10^{-82}$ may be rather large.

For example, the lifetime of the proton (unless it is strictly infinite) is likely to be comparable to $10^{35}$ years which is something like $10^{42}$ seconds. One Planck time, a natural time scale in quantum gravity, is $10^{-43}$ seconds. So the probability that a proton decays in a Planck-time-long interval of time is $10^{-85}$. Relatively to this small number, your number $10^{-82}$ is very large while $10^{-145}$ is negligibly small. At any rate, they're very different.

I could make these tiny values of the probability meaningful because I considered the probability of something happening in a very short period of time – the Planck time – that gets repeated many times in practice. Moreover, I considered a process, the proton decay, that has only affected a tiny fraction of the protons since the Big Bang. Both of these circumstances contributed to the smallness of my number $10^{-85}$. Different processes have different probabilities and different absolute error with which the probabilities may be measured.

Unless I misunderstand something, the actual point of your question is nothing else than the Bayesian-vs-frequentist debates. When the probabilities are considered to be Bayesian measures of subjective, psychological confidence that something is true, of course that they look very fuzzy. If we are making the values of these probabilities ever more accurate by Bayesian inference, such probabilities change all the time, according to rules that aren't quite sharp and that depend on many random and subjective choices. So it makes no sense to express Bayesian probabilities too accurately.

But whenever a probability acquires a frequentist interpretation – whenever it may be measured by repeating an experiment many times – its precision starts to make sense because we may literally measure these probabilities as the percentage of the cases in which the given event occurred. The error is small and goes like $1/\sqrt{N}$.

Both statistical physics and quantum physics predict probabilities that may, at least in principle, be verified by a frequentist procedure. That's especially true for probabilities of lab experiments that take a short enough time. However, even if a physical theory predicts the probability of something that can't be accurately measured in a frequentist way, e.g. the probability that the Universe will decay before the Sun burns its fuel, it doesn't mean that this prediction – and its very precise value – is meaningless. It just means that it's hard to be experimentally verified. We can't repeat the life of the Universe and the Sun many times.

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(i) One can measure probabilities only if they are objective.

A Bayesian only measures events, and adjusts on their basis the prior probabilities. Thus they are not measured but computed on the basis of the events and prior assumptions (i.e., prejudice).

(ii) In a Bayesian interpretation, probabilites are subjectively assigned, and hence can take any real value in [0,1] the subject decides to assign to it. This also includes ideosyncratic values such as for example $1/\sqrt{\pi}$ or even ($1/4$ if the continuum hypothesis is true but $3/4$ if it is false).

It may be ridiculous but is only consistent.

(iii) Subjective probabilities need not be rationally assigned.

Only their change under new evidence is supposed to be rational.

(iv) Even a rational assignment can (and often will) lead to transcendental probabilites if they are the result of a rational mathematical derivation.

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There's a difference between measuring probabilities, and other physical quantities to many significant digits. The fine structure constant in QED has been measured as 7.2973525698(24)×10−3 up to 11 significant digits without requiring $10^{11}$ measurements! By contrast, to measure a probability $1\times 10^{-11}$ requires about $10^{11}$ samples, and a probability of order unity up to the 11th significant digit requires $10^{22}$ samples.

What do you need to do to measure the weight of an object which is 534.6 g with a mechanical triple beam balance? That's a four significant digit precision. First, you adjust the weight for the 100g beam, and note the weight lies between 500g and 600g. Then, you do the same for the second beam and note it lies between 530g and 540g. Then, you slide the slider on the third beam, and note it lies between the long hair markers 534g and 535g. Then, you gently slide the slider a bit each way and look at the fine hair reading, 534.6g. You've just measured the weight without needing to make 10,000 adjustments. You can't do the same thing with probability. This makes probabilities a lot less tangible than weights or other physical quantities.

It's true if you flip a coin 100 times and look at what sequence you get, write down that sequence, and ask what is the a priori probability of getting that sequence, you get $2^{-100}$. However, you get the same answer for a sequence of all heads! If you run statistical randomness tests on both sequences, the all heads sequence will fail practically all of them, while the other sequence, if conducted with tests up to a 95% confidence level, will pass 95% of these tests.

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