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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.
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.
‘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.
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.
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
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
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?
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.
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.
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