TITLE: Probability and Infinity
QUESTION [5 upvotes]: If the probability of an event is $\frac{1}{\infty}$ and $\infty$ trials are conducted, how many times will the event occur — $0$, $1$, or $\infty$?

REPLY [7 votes]: There are meaningful ways to work out probabilities on infinite spaces, but your question is not well-defined.  We could define situations matching your question with the answer 0, 1, $\infty$, or any other finite answer.  
Probability distributions on finite sets are pretty easy to write down and work with.  For rolling a single die, the probability distribution is {p(1)=1/6, p(2)=1/6, p(3)=1/6, p(4)=1/6, p(5)=1/6, p(6)=1/6}.  From this, we can ask and answer many different questions-- for example, the probability of rolling a 3 three times in a row is 1/216.  However, the probability of rolling a 7 is 0; and it doesn't matter how many times you roll, you'll never get a 7.
Probability distributions on infinite sets require a lot more background to define and work with.  You need a probability measure and you can compute integrals to find probabilities.  (Measure theory is usually a graduate-level topic, and it requires very careful, abstract thought to get the details straight.)
An instructive and fun problem to read about is the Random Walk.  Suppose I start at (0,0) on an infinite 2-dimensional grid; and each second, I go up, down, left, or right one unit.  (Each direction has probability 1/4 every second.)  The probability space is the set of all possible walks, which is an infinite set.  The probability measure on this space is determined by the fact that each direction has equal probability every second.
Even though there are many walks in the probability space that never return to (0,0), the subset of walks that do return to (0,0) has measure 1.  This implies that on a 2-dimensional random walk, the expected number of returns to (0,0) is infinite.  However, the expected number of times I take a diagonal step is 0 (because, by my definition of the problem, it never happens).  
We can also consider the 3-dimensional random walk.  Interestingly, on a 3-dimensional random walk, the expected number of returns to (0,0,0) is about 0.516385.  (You can think of this as the average number of returns if you sample many 3-dimensional random walks.)  
These results requires some work to obtain, and you always have to start with a precisely defined problem.  If you think about them and read other examples on your own, it should help you gain some understanding of infinite probability spaces, probability 0 events, and probability 1 events.