Okay, you're right in one sense, because I wasn't completely accurate in my previous comment (I wrote it hurriedly during the last few minutes of my lunch break at work)--the probability does in fact increase somewhat with the sample size. However, your description doesn't describe the situation accurately at all; the correct analogy would be that, each of 100 stars has a 1 in 100 chance of having a planet spiraling into it; in this case, viewing all 100 stars does not yield a probability of 1 of seeing a planet spiraling into it.
Let me try another simplified description. First, consider a single, six-sided die. One of the six sides has a one on it; if you examine a single side of the die (roll it), you have a 1 in 6 chance of seeing the one; if you examine three of the six sides, you have a 1 in 2 chance of seeing the one; if you examine all six sides, you have a 1 in 1 chance (probability of 1) of seeing the one. This is analogous to your "1 out of 100 stars" situation above, but it is not analogous to real life.
Now consider six
normal, six-sided dice. If you roll all six of them, what is the probability that at least one of them will come up with a one? You will probably immediately realize the probability is not 1, but calculating it is a bit of a math problem--it's been a while since my college statistics class, but if I remember correctly, the correct way of finding it is to calculate the probability that none
of the dice will be a one, and then subtract that from 1, thus:
- For each die, the probability of it not being a one is 5/6;
- Thus, the probability of none of the six dice being a one is (5/6)^6, or about 0.335;
- Thus, the probability of at least one of the six dice being a one is 1 - 0.335, or about 0.665, which is significantly less than 1.
Going back to your 1 in 100 probability, if there are 100 stars and each has a 1 in 100 chance of having a planet spiraling into it, then the probability of any of the 100 stars having a planet spiraling into it is 0.634. Examining only 50 of the 100 leaves a probability of 0.5 * 0.634 or only 0.317.
Now, we're making some huge assumptions about the probabilities of this event occurring; but just for the sake of discussion, let's just say that, for any given star, there is a 1 in 10^15 (one in a trillion) chance that, at the present time, it has a planet spiraling into it. (Given the relatively small number of stars we know of that have any planets at all, I suspect that number is a significant overestimate, but I'll use it.) Using your estimate of 100 billion stars in the universe, that makes the probability that any star exists, anywhere in the universe with a planet spiraling into it about 0.0000999, or 1 in 10,000, which is pretty small. Now I'll assume that we have examined 1 billion of those stars closely enough that we would be able to detect this occurrence (which I would guess is a gross overestimate); that makes the probability of any star that we have examined being in the midst of the occurrence about 0.000000999, or 1 in a million.
So, being what I would say is quite generous with all of the numbers, we have a 1 in a million chance of seeing this.