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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 $\infty$?

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1/infinity is not a well-defined probability. Do you mean zero? If so, then it's not possible to say anything; events can occur with probability zero without being impossible. – Qiaochu Yuan Sep 23 '10 at 21:41
Isn’t 1 / ∞ defined as 0? – byoogle Sep 23 '10 at 22:01
Is byoogle pointing out a bug in the reputation algorithm? – yasmar Sep 23 '10 at 22:11
@yasmar: No. If you come over from stackoverflow (by associating accounts), you get 101 rep automatically. – Aryabhata Sep 23 '10 at 22:17
I have started a discussion here about why this thread has so many votes to close it… – anon Sep 24 '10 at 22:15

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.

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Here's one possible answer. Say each event occurs with probability $\varepsilon>0$ and there are $1/\varepsilon$ total events. Taking the limit as $\varepsilon\to0^+$, the event occurs 0 times with probability $1/e\approx36.8\%$, once with probability $1/e\approx36.8\%$, twice with probability $1/2e\approx18.4\%$, three times with probability $1/6e\approx6.13\%$, and so on: the Poisson distribution with $\lambda=1$.

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It's worth emphasizing that this is only one possible answer. The expected number of successes here is 1, but if you have something slightly different like $1/\varepsilon$ trials with $\varepsilon^2$ probability, or $\sqrt{\varepsilon}$ probability, then the expected number of successes will be 0 or $\infty$ respectively. This is an elaboration of Jonas' statement that "we could define situations matching your question with the answer 0, 1, $\infty$, or any other finite answer." – Rahul Sep 24 '10 at 3:40
@Rahul: Absolutely. There are many, many possible interpretations, and I don't want to present this as anything other than that. – Charles Sep 24 '10 at 3:45

Sorry byoogle, I misinterpreted the point of your question. So now I feel compelled to try to answer. I don't know much about probability, but by chance I stumbled across something today that might be interesting to you.

I'll try to frame a question that I think is close to what you are asking, and is also answered by this nice one page paper I found.

So I'm looking for $x \in [0,1]$, and each "instant" I "look", I select a set $A_\alpha \subset [0,1]$. Unfortunately, the probability that $x \in A_\alpha$ is always $0$. But, I have at least this: my set from instant $\alpha$ includes all the previous sets I selected. It seems little consolation though, because at any given $\alpha$ the probability that I have $x \in A_\alpha$ is still 0.

So, if I am allowed to go on looking indefinitely, is the probability that $x$ is "found" zero? Well, it seems that the probability can in fact be 1, if we choose well. At least that is how I interpret this:
When you read that, you can read "measure zero" or "negligible" as "probability zero".

There is a fair chance that my analogy is broken, and hopefully that will be pointed out if that's the case. Our "instants" can't be countable, in other words we need to be continuously looking, rather than just looking every microsecond or so. Also, my analogy seems to be using the ordering on the real numbers, and maybe that is a logical stumble.

I hope that helps, or is at least amusing.

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You're probably missing a property on A_a that it is sufficiently small. At this point A_a can be [0,1) and I don't think the probability is 0 for that subset. Also, I don't think "looking indefinitely" is the same as taking the union like in the PDF. For example, I can "look at" sets of numbers, starting with {0} and every time adding max+1. How ever long I "look" the sets remain finite, yet the union of all sets (which I've seen and will see if I continue indefinitely) is the set of natural numbers and infinite. – mweerden Sep 24 '10 at 1:22
@mweerden Thanks for mentioning these things. The picture I was trying to paint is that x is unknown. You are alowed to pick any $A_\alpha$ you want, provided it has measure 0 and includes your previous choices. This means the probability that $x$ is in your set is zero. Of course you can make bad choices and never get a positive probability, but the point is that it is possible to select such sets in a way that the probability "at eternity" is positive, and even 1. – yasmar Sep 24 '10 at 1:36
"measure 0" is the sufficiently small part. If the set has positive measure, the probability that it contains the unknown x is no longer zero. – yasmar Sep 24 '10 at 1:38

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