Let's say there is a 10% chance of an event taking place per 100 samples. If you increase the number of samples to infinity, theoretically the chance of the event occurring would converge 100%. But imagine this:
If you treat infinity as an infinite amount of sets of 100, would each of those sets still have only 10% chance of the event taking place?
Yes on both accounts...
If there is a 10% chance of an event taking place per 100 samples, then then the probability of an event taking place is $p = 0.001$ per sample. Then let $X_n$ be the number of events that occurs in a sample of $n$ trials. Clearly,
\begin{equation} X_n \sim Binomial(n,0.001)$. \end{equation}
To see the first question, consider:
\begin{equation} \lim_{n\rightarrow\infty}P(X_n = 0) = \lim_{n\rightarrow\infty}0.999^n = 0 \end{equation}
To answer the second question, let $Y_1, Y_2, ... \sim Binomial(100,.001)$
\begin{equation} \lim_{n \rightarrow\infty}P(Y_n > 0) = .001 \end{equation}