Let's assume there are an infinite number of planets.
Similarly, given that we exist, we know that the probability of life emerging on a given planet is strictly greater than 0.
Assuming that by know here you mean that it directly follows, then surprisingly enough, no, we don't know that.
Let's go to the marbles and I'll explain why:
Suppose we have an urn that contains infinitely many balls with different colours. Black balls are very common, and white ones uncommon.
I propose a minor change; let's say the balls are equally common. We draw a black ball with probability
1/2, and a white ball with probability
1/2, those being collectively exhaustive and mutually exclusive, regardless of the number of draws.
Now let's draw an infinite number of marbles. Since I imagine doing so can get pretty boring, I propose we play a game as we do so. I'll supply you with one infinitely dense chocolate bar, labeled along its length equally with all of the real numbers from 0 to 1. I'll draw the marbles. Before each draw, you cut the chocolate bar exactly in half along the length. After the draw, you get to eat one of the halves; if I draw a black marble, you eat the larger numbered half (leaving the lower one for the next turn). If I draw a white marble, you eat the smaller numbered half.
To give you an idea; after three draws, you will have eaten
7/8 of the chocolate bar. There will be
1/8 remaining, but there are
8 different ways this can happen. I propose a simple observation; after some number of draws, you will have eaten some amount of the chocolate. However long that uneaten bar is, the a priori probability you will have wound up with that specific portion of chocolate is the same as its length.
We're now ready to play the game an "infinite" number of times, somehow. By doing so we'll wind up inching towards some point-thin slice of uneaten chocolate labeled with some real number. That slice of chocolate necessarily has a zero width (in particular, were I to challenge you to name any positive width, no matter how small; I can tell you the finite number of draws that sliced the bar smaller than that).
Now note my previous observation; the length of the slice of uneaten chocolate is the same as the probability that that particular piece would be selected. The length of your point thin slice is
0. So the probability that it would get selected is
0. But we will select some piece by playing this game; it's impossible not to. It follows, then, that there's something counterintuitive going on with probability once the infinite gets involved; there's such a thing as an event that has a probability of exactly
0, but is nevertheless possible. Formally, an event that has a probability of
0 but can still occur is said to "almost never" occur, which is distinct from being impossible (though impossible events also have probability
0). The complement (an event that has probability
1 but can still not occur) is said to "almost surely" occur, which is distinct from being certain (though certain events also have probability
So let's backtrack now. Suppose there's a probability
1/2 on each planet that life will be there (a "pure" probability, like "flipping a coin" or more aptly "drawing marbles from an infinite urn"; not a frequentist one). Then it's no surprise we're here; but given the infinite number of other planets, almost surely some other planet will have life on it. But it's still possible none of them do (this is equivalent to selecting
1/2 on the bar; i.e., the first time you pick a white marble, then never do again).
So in summary, given life exists on our planet, but under the assumption that there are an infinite number of planets (or even not strictly), all we can establish with certainty is that life is possible (that is, we're here, therefore it's possible we're here). That doesn't even imply that the probability of a planet having life is greater than
0. But even if the probability were greater than
0, we cannot be certain there's life elsewhere... even if that probability is high; at best we can claim that there's almost surely alien life.
That's still a lot; I wouldn't bet against almost surely.
(FYI, the above maintains a non-Bayesian view of probability; using a Bayesian approach would clash with the proposed notion that we don't know what the probability is).