There is no reason it should.

Eta: is this a well known issue in statistics or philosophy of mathematics?

It's not clear what your question means.

If it means "why does a dice land with any given face upwards one sixth of the time?" or something akin to that, then the answer is that it might, or it might not. Probably it doesn't.

Statistical modelling, like any mathematical modelling, scientific modelling or logical modelling depends on making assumptions, and fitting those assumptions to real physical problems. The better the assumptions fit, the better the model fits and the more reliable its predictions are. The benefit of assumptions is that they allow you to use sophisticated derived results.

Perhaps on a broader level we could say that the answer to "why does probability work?" is essentially the same as "why does Euclidean geometry work?" or "why does Newtonian gravitation work?". If the assumptions are good enough for the situation, then it works, if they aren't, then it doesn't, which in those latter examples would be the case for sufficiently large scales.

I was going to say something similar -

- why do Newton's laws of motion operate? and the laws of thermodynamics: what's that about? -

but Thommo has given a more complete answer.

Maybe other, alternative laws might make sense too (and there used to be other ideas about the laws governing motion, for instance); but the fact is (or seems to be) that these laws, including the laws of probability, actually operate in the reality that we are observing.

If you want a philosophical treatment of interpretations of probability (which doesn't seem to really be the question, but might be tangentially relevant or illuminating), you could try something like this:
http://www.siue.edu/~evailat/interp-prob.htm

The main interpretations of probability are best divided into into two groups:

• Epistemological interpretations, according to which probability is primarily related to human knowledge or belief.
  
• Objective interpretations, according to which probability is about a feature of reality independent of human knowledge or belief. Sometimes reality is taken to be the physical world; at times it is taken to include a sort of Platonic realm of mathematical and logical entities.

Or this:
https://plato.stanford.edu/entries/dutch-book/

The conclusion of the DBA is that the degrees of belief, or credences, that an agent attaches to the members of a set X of sentences, statements, or propositions, should satisfy the axioms of probability.

(be warned, there's an economist's "should" in that sentence)

Or this:
https://plato.stanford.edu/entries/prob ... interpret/

‘Interpreting probability’ is a commonly used but misleading characterization of a worthy enterprise. The so-called ‘interpretations of probability’ would be better called ‘analyses of various concepts of probability’, and ‘interpreting probability’ is the task of providing such analyses. Or perhaps better still, if our goal is to transform inexact concepts of probability familiar to ordinary folk into exact ones suitable for philosophical and scientific theorizing, then the task may be one of ‘explication’ in the sense of Carnap (1950). Normally, we speak of interpreting a formal system, that is, attaching familiar meanings to the primitive terms in its axioms and theorems, usually with an eye to turning them into true statements about some subject of interest. However, there is no single formal system that is ‘probability’, but rather a host of such systems. To be sure, Kolmogorov's axiomatization, which we will present shortly, has achieved the status of orthodoxy, and it is typically what philosophers have in mind when they think of ‘probability theory’. Nevertheless, several of the leading ‘interpretations of probability’ fail to satisfy all of Kolmogorov's axioms, yet they have not lost their title for that.

I can't say that (aside from some aspects of the Dutch book arguments) they do a lot for me, but maybe that's what you want.

I had some confidence that this question would be asked before the year ended but there was no surety and I didn't place a bet.

Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other

Just to be clear, probability theory does not say a fair coin tossed 100 times will land 50/50 heads and tails. Consequently if a coin lands 51/49 it does not entail the conclusion that one side is marginally heavier than the other, that would be an example of The Gambler's Fallacy.

newolder wrote:I had some confidence that this question would be asked before the year ended but there was no surety and I didn't place a bet.

Can you name a pointless answer?

surreptitious57 wrote:Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other

But these are all tautological answers. My question is why does the coin flip vary around a bell curve? Why wouldn't it be always heads for no reason at all?

surreptitious57 wrote:Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other

The first part is tautological, probability works because it works. But the second part gets at my question kind of. If it was 51-49, that wouldn't be statistically significant but 80-20 would definitely indicate that there was a missing variable. Why?

Don't know if this helps, but just to narrow down the enquiry: the question and its answer or answers relate to science, not to mathematics. Mathematics just defines the functions that describe the observed behaviour.

So it's not a mathematical law that says that probability works; it's a scientific observation. It seems to make sense (Why a bell curve? Because when a coin flips, there is no reason why it should come up heads rather than tails, so we can expect it to do both about equally often), but that's an intuitive, emotional, and - as you say - tautological argument equivalent to saying that Newton's laws of motion apply because they're common sense.

It only works a percentage of the time.

Heh. The jokes almost write themselves. But yeah. it's a related issue to the unreasonable effectiveness of mathematics in the natural sciences.

If probability theory didn’t work, we wouldn’t use it.

The OP’s question seems to be Hume’s problem of Induction rephrased.

BWE wrote:

surreptitious57 wrote:Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other

But these are all tautological answers. My question is why does the coin flip vary around a bell curve? Why wouldn't it be always heads for no reason at all?

Coin flips are not distributed by a bell curve (AKA Normal distribution or Gaussian distribution), they are binomial distributions.

There's an important theorem in mathematics that governs the distribution of sample means from a wide range of arbitrary distributions, known as the https://en.wikipedia.org/wiki/Central_limit_theorem, which says that for large sample sizes the distribution of sample means is approximately normal regardless of the shape of the original distributions.

In terms of the question "Why wouldn't it be always heads for no reason at all?" it's poorly phrased. A series of coin flips of a fair coin of arbitrary length might be all heads, probability theory does not forbid this. And similarly a coin might be biased (such as by having two heads) such that it only ever flips heads. However in this latter situation the assumption that the coin is fair does not hold, and a model is only as good as its assumptions.

scott1328 wrote:If probability theory didn’t work, we wouldn’t use it.

The OP’s question seems to be Hume’s problem of Induction rephrased.

Maybe. But it does seem like there's a deeper issue going on. There's the brute fact aspect involving observation but there's some kind of modeling mystery in there too. At least for me.

On the bell curve, I interpreted the statement as referring to a large number of experiments, each consisting of a certain number of coin flips: the results of the experiments would cluster around 50-50 in the shape of a normal distribution.

Thommo wrote:

BWE wrote:

surreptitious57 wrote:Probability works because it attempts to model reality based on observation
Every probability exists between 0 and I and how accurate a prediction will
be is determined by the accuracy and completeness of the data in question

You toss a coin I00 times probability says it will land heads 50 times and tails 50 times
Reality however says it is 5I / 49 because one side is marginally heavier than the other

But these are all tautological answers. My question is why does the coin flip vary around a bell curve? Why wouldn't it be always heads for no reason at all?

Coin flips are not distributed by a bell curve (AKA Normal distribution or Gaussian distribution), they are binomial distributions.

There's an important theorem in mathematics that governs the distribution of sample means from a wide range of arbitrary distributions, known as the https://en.wikipedia.org/wiki/Central_limit_theorem, which says that for large sample sizes the distribution of sample means is approximately normal regardless of the shape of the original distributions.

I think you understood my point though.