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How do we prove the probability of any event would converge into some constant value, for infinite number of trials?

                          P(E) = k
                       where E -> An event
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closed as not a real question by TMM, tomasz, Alexander Gruber, Asaf Karagila, Chris Eagle Dec 30 '12 at 16:47

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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That's a very vague question. Can you give an example? –  Thomas Andrews Dec 30 '12 at 14:08
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I don't think we do. We usually assume that the probability is the same for each trial. Then the sequence of probabilities is the constant sequence, which of course is convergent, but that is not a very illuminating fact. –  Henning Makholm Dec 30 '12 at 14:09
    
@ThomasAndrews : I have edited it. I was not asking for any particular event. –  Inquisitive Dec 30 '12 at 14:10
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I think he might be looking for the proof of the Law of Large Numbers. –  Amzoti Dec 30 '12 at 14:11
    
@HenningMakholm - If the probability is same for each trial, then would that sequence converge? I don't think so. –  Inquisitive Dec 30 '12 at 14:12

1 Answer 1

up vote 2 down vote accepted

I think what you're really trying to ask is:

If we assume that the probability of some event is the same for each trial, how do we prove that the observed frequency will tend towards that probability?

That is (a version of) the law of large numbers, and it holds only with overwhelming probability, not with absolute certainty.

The raw fact that the observed frequency will tend to the probability is more of an axiomatic assumption (or an empirical fact) than a derived result. What the technical statements of the law of large numbers will give you is precise values for how unlikely it is for it not to be close to the probability. This quickly becomes very unlikely, which depending on how you look at it is either profound or trivial, because it is just probability theory predicting that its own basic assumptions are overwhelmingly likely to be true.

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+1 especially for that last sentence. –  Dilip Sarwate Dec 30 '12 at 14:24

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