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In an experiment such as tossing an unbiased coin, we know that the probability of getting a 'head' or a 'tail' is 1/2. I understand this to mean if the number of trials is large enough. My question is how is that "large enough" number determined mathematically? Is there a formula for it? I can guess it would have to depend on the sample space, I mean for a dice with 6 faces, that number should be different from the coin of 2 faces. I am sure also that it is at larger or equal the sample space but I want to know the formula for it if such a formula exists.

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    $\begingroup$ Is your question: hoe many times do you have to throw a fair coin to see them occurring with frequency exactly equal to 1/2 ? If it is so, then the answer is: infinite number of times. No theory states that there is some finite sample size that will produce such outcome. $\endgroup$
    – Tim
    Commented Nov 7, 2016 at 13:26
  • $\begingroup$ @Tim, yes that is my question exactly. I am surprised with your answer though! Could you suggest any references? Thanks. $\endgroup$
    – NoChance
    Commented Nov 7, 2016 at 13:40
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    $\begingroup$ In applied statistics the question of "large enough sample" is usually asked in context with a specific statistical test. The name of that context and hence the search term for you favourite search engine is "power analysis". You seem to head towards a binomial trial, so "power analysis binomial" could be a good search phrase. A free software package especially for practical power analysis is called G*Power. Hope, any of this helps. $\endgroup$
    – Bernhard
    Commented Nov 7, 2016 at 13:41

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By the weak law of large numbers

$$ \lim_{n\to\infty}\Pr\!\left(\,|\bar{X}_n-\mu| > \varepsilon\,\right) = 0 $$

so as your sample grows up to infinity, the empirical mean $\bar{X}_n$ will get closer and closer to the true mean $\mu$. And by the strong law of large numbers, as sample size goes to infinity, the empirical mean will converge almost surely to the true mean

$$ \Pr\!\left( \lim_{n\to\infty}\bar{X}_n = \mu \right) = 1 $$

As you can see, neither of the statements say that there is any finite sample size that will let you achieve this. On another hand, they do not say that there is some specific sample size that is needed to observe some event with some probability: if you throw a fair coin only once, the probability of heads is still $1/2$.

The thing that you could estimate is the standard deviation of your estimate of probability of heads $p=1/2$ after $n$ trials using Wald's formula

$$ \sigma = \sqrt{ p(1-p)/n } = \sqrt{0.25/n}$$

so you could find $n$ large enough for $\sigma$ to be small enough for you to consider the variability of $p$ acceptable. But still, this is your subjective choice of what do you consider as "small enough".

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  • $\begingroup$ This is great. It is so kind of you to take the time to teach me this. Thank you very much. Many of the text books I looked at had nothing to say but that the probability of head or tail is 1/2. $\endgroup$
    – NoChance
    Commented Nov 7, 2016 at 14:00

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