What is the intuition behind the Borel Cantelli Lemma? The Borel Cantelli Lemma states that the probability of a set of outcomes happening infinitely many times in an infinite sequence is $0$ under some conditions. I neither understand the math (behind the proof) nor the intuition behind what is being said.  Can someone motivate the use of this theorem?

Toss a coin with $X_1, X_2,...$ denoting sequence of independent Bernoulli trials and probability of success (head or tail or whatever) on $n^{th}$ trial be $p_n$. Then Borel- Cantelli lemma tries to answer what is the probability of an infinite number of successes, i.e. $P(X_n = 1 \ i.o$). This, is either zero or one depending on if $\sum p_n < \infty $. So if you choose your $p_n$ judiciously, for example, if $p_n=1/n^2$, then $P(X_n=1 \ i.o)=0$. Similarly, $P(X_n=1 \ i.o)=1$ if $p_n=1/n$.
