'Probability method' - to what extent is it an actual proof? Consider this: > $\frac { C^{2n}_n } {2^{2n}} = \mathbb P (A) $ where A= { equal number of heads and tails in $2n$ throws of a fair coin } > > therefore the following assertion is true: > > $\forall n \ge 0 : \ \ 0 \le \frac { C^{2n}_n } {2^{2n}} \le 1$ my question is: To what extent is this a bona-fide 'proof' of the assertion, if at all?

It is a perfectly valid argument. If you know that the formula corresponds to a probability, the value must be between $0$ and $1$, because all probabilities are. However, as a "proof" it lacks bona-fides in the sense that a proof proceeds from commonly-accepted premises (axioms) to less-obvious conclusions (theorems), and here you are working essentially the opposite way, since the validity of the formula is much less obvious (to most people) than the fact that a probability must be between $0$ and $1$ (which is in fact one of the probability axioms). That said, there is nothing wrong with reasoning this way. In the right context it could be really slick. One thing to watch out for is cases when the formula isn't valid as a probability, for example negative values of $n$.

You may be interested in Erdos' probabilistic method which is basically a way of showing something exists because its probability is non-zero.