TITLE: How do concepts such as limits work in probability theory, as opposed to calculus?
QUESTION [17 upvotes]: When I am flipping a fair coin and say that as the number of trials approaches $\infty$ the number of heads approaches $50\%$, what do I really mean?
Intuitively, I would associate it with the concept of a limit, as used in calculus:
$$
\lim_{t \to \infty} \left(\frac{H}{H+T}\right)=0.5 \\
\text{Where $t$ = the number of trials, $H$ = the number of heads, and $T$ = the number of tails}
$$
However, this intuition seems to break down when I use the formal definition of a limit:
$$
\lim_{t \to \infty} \left(\frac{H}{H+T}\right)=0.5 \text{ if and only if}\\
\text{for every $\varepsilon>0$, there exists $N>0$ such that for all $t$} \\
\text{if $t>N$ then $|0.5-\frac{H}{H+T}|<\varepsilon$}
$$
Well I don't know if there will be $N > 0$ that satisifes this definition! It all depends on what comes up. However many times I flip the coin, $\frac{H}{H+T}$ might just equal $0$. So how might I define terms such as "approaches" in probability theory if the conventional definition does not work?
Edit: User nicomezi has pointed out that this a huge topic. Therefore, I will accept even a very short introduction to this subject as an answer.

REPLY [3 votes]: In probability, you work with distributions. When you increase the number of trials and take the average, you get a distribution that tends to a Gaussian, with a narrower variance.
The conclusions that you can draw are based on this probabilistic inference. For an infinite number of drawings, the distribution tends to a Dirac delta.