I would like to have a better intuitive feeling concerning the difference between the two modes of convergence. Consider the following example :
Let $(X_n)_{n \geq 1}$ be a sequence of independent Bernoulli random variables such that $P(X_n=1) = p_n \rightarrow 0$.
Depending on the speed of convergence of the $p_n$, the sequence will converge a.s. or only in probability. Take $p_n = \frac{1}{n^2}$ or $p_n = \frac{1}{n}$ for instance. On one side, the event $\{X_n=1\}$ has positive measure for all $n$ in both cases. On the other side the Borel-Cantelli lemma and the "converse result" (legal since the $X_n$ are independent) can be used to directly prove that the sequence is $0$ a.s. after a certain rank when $p_n = \frac{1}{n^2}$, but infinitely many $1's$ are expected when $p_n = \frac{1}{n}$.
What causes intuitively those two different behaviors?