How many experimental trials are needed to invalidate a theoretical probability? I became curious of this question after watching this video:
https://www.khanacademy.org/math/statistics-probability/probability-library/experimental-probability-lib/v/comparing-theoretical-to-experimental-probabilites
It says that if we have a 1/2 probability of extracting a blue marble or a yellow marble, if we do the experiment enough times we should have a 50% ratio.
But if we do 10 experiments and we have 7/3 ratio is normal.
However if we do the experiment 10,000 times and we have 7000/3000 then it is not normal.
How do we know when an experiment is contradicting or validating a theoretical probability?
 A: Before introducing a quantitative answer, let one say that when you are dealing with realisation of a random variable, the randomness is structured by the distribution of probability.
Here you are dealing with a Bernoulli (some will say binomial) of unknown parameter $p$, and you want to estimate $p$ from several realisations.
Here, the quantity 
$$\sum_{n=1}^N \frac{1}{N} 1_{success}(X_n), $$
where N is the number of realisation and $X_n$ the realisations, converge in a sense to $\mathbb{E}[X_1]=p$.
You Computing $7/10$ or $7000/10000$ you are computing an estimator of $p$.
The rate of the convergence is given by a central limit theorem and is $\sqrt{N}$. So the more the number of sample, the better you estimator will be.
I guess the best way to measure whether a random phenomenon is normal or not is to have a look at confidence interval.
In a few words, confidence intervals tell you how likely an event will occur.
Indeed, as soon as a positive probability is attributed to an event, you can always find one or more $\omega$ such that this event will occur.
However, the number of $\omega$, or equivalently the measure of the set containing those $\omega$ may be small.
Confidence intervals furnish an answer to that.
Computing confidence intervals can barely be done exactly, but lots of research has been pursued to approximate them.
Since you are dealing with Bernoulli event, a classical confidence interval is the clipper-peason one.
You can easily compute it with python, following for instance this code.
The clopper-pearson confidence interval takes three arguments : the number of trials, $n$, the number of success, $k$ and the confidence interval size $\alpha$.
For instance, for $\alpha=0.01$, it will provides a segmente $[a, b]$ where $a$ and $b$ are such for $1-\alpha=90\%$, the true value of the parameter will be in $[a, b]$.
Here, for 10 trials and 7 successes, a 99% confidence interval says that the parameter is in [0.26, 0.96], so it is a plausible realisation for $p=0.5$.
For 10000 trials and 7000 success, it gives you a confidence interval of [0.69, 0.71].
