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When a random experiment results in a particular outcome we beleive it could have resulted in some other possible outcome as well. Consider, for example, an experiment of flipping a coin which has come up heads.What does it mean to say that it could have come up tails as well? Which experiment on earth can empirically show that it was possible for the coin to have come up tails on that very trial? Of course it can come up tails in some other trial but that is irrevelent to our trial which has resulted in the head. I am not criticising the theory of probability; I just want to know if the assumption of "could have happened otherwise" is a priori and not empirically refutable or verifiable!

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If it is a soft question, you may tag it as such. –  Jeremy Jan 4 at 5:57
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Understanding the meaning of probability is surprisingly subtle. One viewpoint is that probability is simply a measure of how much one believes something. The book Probability Theory: The Logic of Science by Jaynes presents this viewpoint. –  littleO Jan 4 at 7:19

4 Answers 4

What does it mean to say that it could have come up tails as well?

It means that we must know a priori the possible outcomes of an experiment (or that we must believe, or assume, that we know them). This knowledge, or this belief, or this assumption, is a pre-requisite in order to discuss probabilities, in a theoretically consistent manner.

Which experiment on earth can empirically show that it was possible for the coin to have come up tails on that very trial?

An experiment for which one could show that it was conducted under the exact same circumstances (those subset of "circumstances" that are relevant to the outcome) with the "very trial under consideration", and one in which the coin came up tails.

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Not sure I completely understand the question, but the idea of the set of all possible outcomes is simply the sample space from which you are sampling. In your example, the sample space would be $\{\text{heads}, \text{tails}\}$. These are simply the measurable sets in your space. Any of them could occur, but only some element from the all events will. The probability of any event occurring is the value of that measure of that set.

See this article for more info: http://en.wikipedia.org/wiki/Probability_space

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It's a great question. The a priori assumptions are as follows.

1) given repetition with all the same precise parameters, the outcome would be the same. (i.e., there is no reason to discount determinism: robots in controlled environments can replicate effectively perfect coin flips. Also, classical mechanics is effectively accurate without appeal to quantum mechanics.)

2) the parameters are too variable, and the outcome too dependent on minor perturbations to be discernible to an untrained human. (i.e., you'd expect temperature fluctuations, barometric pressure, physiological parameters like glucose, blood pressure, strength, etc. to all affect the outcome). None of these parameters has any reason to favor H or T a priori. (Trained humans, such as at least one of the authors of the below citation can achieve much more deliberate results).

One would like--and so one's model insists--that the parameter space is checkered evenly with outcome H and outcome T. I.e., the measure of T events, subject to the flipper's particular probability distribution over the parameter space is equal to the measure of H events over the same. In reality, whatever side is up to begin with has a slight advantage to landing up. This bias is shown in http://comptop.stanford.edu/u/preprints/heads.pdf . However, your question seems more about any physical process expected to be random and not about coins in particular.

The final thing to say is that probability is in part in the eye of the appraiser, since from 2 above, a keen enough discerning robot could tell you the outcome as the coin is leaving your hand. This is the scenario for cards, where there is never a chance that the top card is any other than what it is. But as is the common scenario, you have no reason to favor it being any one of 52 cards over another.

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This isn't really a mathematical question. Probability theory works by assuming that there's some set of possible outcomes, and that numbers can be assigned to certain sets of outcomes. As to why this models events in the "real world," that's a philosophical question.

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Of course I am not saying it is a mathetmatical question. i am primarily interested in mathematics but one cannot just explain away profound philosophical questions about mathematics merely by retorting that they are philosophical.And it is often disappointing to expect good aswers of them from philosophers with no or superficial knowledge about mathematics .I mean if we seriously care about methemtics then we must seriously care about its philosophical implications and foundations as well;we cannot simply relegate those questions to philosophers! –  sajjad veeri Jan 4 at 8:17
    
By "philosophical" I mean "unanswerable." –  Daniel McLaury Jan 4 at 8:26

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