I encountered the following two questions in a measure theory class: (1) Suppose that I flip a coin many times, and the probability that the coin turns heads on the $n^{th}$ flip is $\frac{1}{n}$. Will the coin ever stop flipping heads? (2) The same setup as above, but the probability that the coin turns heads on the $n^{th}$ flip is $\frac{1}{n^{1+\varepsilon}}$, where $\varepsilon > 0$. Will the coin ever stop flipping heads?
I know that by the Borel-Cantelli Lemma, $\sum_{n=1}^{\infty} P(A_n) = \infty \implies P(A_n \text{ i.o}) = 1$ and $\sum_{n=1}^{\infty} P(A_n) < \infty \implies P(A_n \text{ i.o}) = 0$ (where in the former the events must be independent). Thus, the answer to (1) is no but to (2) it is yes (given the divergence of $\frac{1}{n}$ but the convergence of $\frac{1}{n^{1+\varepsilon}}$).
I totally get why this is the case based on application of the theorem. But I am having trouble thinking of it intuitively. Why does even the smallest change in the probability of heads make all the difference in saying that, at some point, the coin will stop flipping heads? Does anyone have an intuitive way to think about this?
Thanks!