TITLE: What is the intuitive difference between almost sure convergence and convergence in probability?
QUESTION [11 upvotes]: It is a standard fact in probability that almost sure convergence is stronger than convergence in probability. I can only see the differences in the proof. However, is there a way to view it intuitively? Is it true that almost sure convergence has a tighter hold on the tails of a sequence of random variables than convergence in probability does? The definition of convergence in probability I am using is that given $\epsilon >0$:
$$
\lim_{n \to \infty} P(|X_n-X|> \epsilon) = 0
$$
and the definition of almost sure convergence I am using is:
$$
P(\lim_{n \to \infty}X_n = X) = P\left(\omega \in \Omega: \lim_{n \to \infty}X_n(\omega) = X(\omega)\right) = 1
$$
The two above appear almost exactly the same to me, except that the limit on $n$ is outside of convergence in probability and within the probability measure for almost sure convergence. Is there an easy to understand intuitive difference here? Thanks!

REPLY [7 votes]: For simplicity, consider the case where $X = 0$ and $X_n$ is the indicator function of an event $E_n$.  "$X_n$ converges almost surely to $0$" says that with probability $1$, only finitely many of the events $E_n$ occur.
"$X_n$ converges in probability to $0$" says that the probability of event $E_n$ goes to $0$ as $n \to \infty$.  
Consider a case where for each $m$ you partition the sample space into $m$ events, each of probability $1/m$, and take all these events for all $m$ to form your sequence $E_n$.  Then $X_n \to 0$ in probability because the probabilities of the individual events go to $0$, but each sample point is in infinitely many 
$E_n$ (one for each $m$) so $X_n$ does not go to $0$ almost surely.