# Rate of convergence (Probabilty)

I didn't take any probability classes and just remember a bit about what i did in Highschool. This might be a very elementary question but i'm really curious about it.

Anyway here goes my question:

denote by $N(i)$ the number of times that the experiment turnt up the $i$'th possibility. denote by $N$ the total number of expremients (or trials). denote by $P(i)$ the probability of the $i$'th possibility. Then we have : $\frac{N(i) }{N }=P(i)$ as $N \to \infty$.

Now i would like to find a simple example of an experiment so that we can determine a $m$ and an $\epsilon$ for wich we have :

$$\forall N\geq m \qquad \left|\frac{N(i) }{N }-P(i)\right|<\epsilon$$

If the question is not clear i will try to restate it. Thanks in advance.

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It is possible that if you toss a fair coin a very large number of times, you will get all heads. So the assertion needs to be modified. For details, please see Wikipedia, Law of Large Numbers. – André Nicolas Jul 11 '13 at 23:04
The usual notion of a limit is actually not applicable to convergence when it comes to probability. Check out the following for what we mean when finite experiments "converge" probabilistically. en.m.wikipedia.org/wiki/Convergence_of_random_variables – rajb245 Jul 12 '13 at 1:57
i knew that my thoughts where too simple...thanks for the suggestions! – sigmatau Jul 12 '13 at 2:01