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Trying to understand the basic idea behind probability: If you have a dice, and you throw it an infinite number of times, then the probability of getting each side of the dice is 1/6. So far logical. Now, this predicate "infinite number of times" is what disturbs me. Is the science of probability all based on this unrealistic assumption?

If we were to transform the problem to that of life and death. What is a probability that a person will die in the next second? Answer is 1/2 : either yes, either no. What if the person has an incurable disease, what is the probability of the person to die in the next second? Answer is still 1/2 : either yes, either no. OR Answer is 1 / (estimated quantity of remaining time to live?). Because simply there is no way to repeat the experiment an infinite number of times, both persons have equal probability of living!!

Excuse me for this weird question,thanks you very much for your time!

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The probability of getting one particular side on a fair die in a single throw is $1/6$ (among other things because that's the definition of "fair die"); you don't need to "throw it an infinite number of times." But if you actually throw a die, say, 600 times, you will probably not get exactly 100 times an outcome of 1, exactly 100 outcomes of 2, etc. Instead, there will be a small variation between the expected probability (1/6 for each side) and the actual observed outcomes. What you expect is that discrepancy to diminish as the number of rolls increases. –  Arturo Magidin Oct 31 '11 at 18:27
    
Thanks. OK, so there is an assumption of 'fairness', and the number of experiment has to reach a meaningful threshold. These two predicates are essential! –  k.honsali Oct 31 '11 at 18:38
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I certainly hope the probability of my dying in the next second is a lot less than $1/2$. The mere fact that there are two possible outcomes does not make them equally likely. There are real philosophical difficulties in speaking of probabilities of events that are not repeatable. See en.wikipedia.org/wiki/Probability_interpretations –  Robert Israel Oct 31 '11 at 18:43
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@k.honsali: No: the "assumption of fairness" is just that there are dice that are loaded; you don't expect those dice to behave the same as regular dice. (Do you expect a two-headed coin to fall heads half the time, and tails the other half?) As to "number of experiments has to reach a meaningful threshold"... a "meaningful threshold" for what? Not for probability to work. That's not it at all. You seem to be confusing a specific observations with probability. If you toss a coin and it falls heads, then it's been heads 100% of the time. Does that disprove the theory of probability? –  Arturo Magidin Oct 31 '11 at 18:44
    
@k.honsali: Your second paragraph is also incorrect. Just because there are two possible outcomes does not mean each outcome is equally likely. Is there life on Mars? 50% probability yes, 50% probability no. Is there intelligent life on Mars? 50% probabilty yes, 50% probability no? That would mean that you have the same probability of having life and of having intelligent life, which is certainly not the case. –  Arturo Magidin Oct 31 '11 at 18:46
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It's not based on that irrealistic assumption. The central limit theorem states just that you cover your intuition of probability of an event at infinity as an actual event: indeed, when talking about dice you have a complete knowledge about what is happening: you are assuming that the dice is non-loaded, so that any side has probability $\frac{1}{6}$. The central limit theorem tells you that you get that probability at infinity.

If you talk about life and death, you don't have any real way to estimate (non trivially) the probability to die the next second.

I think that the theory of probability is, from a philophical point of view, to give a certain value to an uncertain event, describing its quantity of uncertainty. But how can you do that if you don't know the quantity of uncertainty?

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Thanks> Nice! OK, I understand that the Central Limit Theorem proves that the P. of an event at infinity corresponds to the P. of the actual event at every one time. For a gambler, say, who throws at most 100 dice a day, is it safe to extrapolate those results at infinity on the actual throw of dice? Maybe this is the only way to give a 'logical/objective' estimate of P.. The problem of life and death was used to illustrate how relevant is the estimation to the actor who actually experiences the event. If I may reason like this... –  k.honsali Oct 31 '11 at 18:58
    
What does "extrapolate those results at infinity on the actual throw of dice" mean? And what does "safe" mean? If you mean, "can the gambler assume that on exactly 1/6th of the throws he will get 1, on exactly 1/6th he will get 2, etc" then of course not (and it has nothing to do with the fact that 100 is not divisible by 6). What he can assume is that, on average, he can expect a particular outcome to occur about 1/6th of the time, with the "about" being closer and closer to exactly 1/6th the more throws he makes. That does not preclude "good/bad runs" from ocurring, though. –  Arturo Magidin Oct 31 '11 at 19:17
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Part of your question has to do with the statisticians' debate over the frequentist vs. Bayesian interpretation of probability. This link here seems to have a fair explanation.

The second half of your question seems to make the mistaken assumption that, for all questions (such as will I die in the next second) the number of possibilities (2, yes or no) have equal probability. Clearly that's not true. There are two possibilities, but this is not a coin toss. Perhaps there are any number of freak accidents that could take me out in the next second, and I don't actually know if they're going to happen; so I have to give some probability to that eventuality, but the probability is quite small. This is the uncertainty/Bayesian interpretation. From a frequentist point of view, one could think of multiple time lines sprouting from the present reality--if those timelines represent all possible things that could happen (assuming that the future is actually occurring probabilistically, and not according to chaotic determinism), given the present circumstances, then perhaps, in some of those timelines, I die pre-maturely. But I sort of prefer the Bayesian interpretation in this case.

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Interesting, I understand the inequality of possibilities, and the Frequentist/Bayesian paradigm.. Next, how would the Bayesian interpretation formulate the equation for life? How would it model/program such a simulation? I see an ANN, with nodes representing the frequency/probability and inputs the eventuality/probability, the output could be propagated back to represent history. –  k.honsali Oct 31 '11 at 21:48
    
@k.honsali I'm not sure what you mean by 'the equation for life'. No one would attempt to model something complex as a whole life in any complete sense. It's just intractable. To model something like timing/probability/cause of death, a Bayesian would look at actuarial tables to find the ways and ages that people die as well as any additional data about the person. The Bayesian could then take all known conditions of a person of interest and compare them to the conditions of people that died and come up with a probability of the person's death. They'd work for an insurance company. –  JCooper Nov 1 '11 at 3:11
    
thanks @JCooper, I was just curious to push the limits of my understanding to a 'practical' implementation.. –  k.honsali Nov 2 '11 at 16:35
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