I find myself embroiled in debunking some of the unfounded claims that the LHC will somehow destroy the universe. Perhaps you've heard of this, the argument usually involves Black Holes, Strangelets, or the newest one, Bose-Nova's.
The fundamental logic of these claims is flawed, and let me explain how. Generally these people want some kind of 100% proof that the LHC is safe. Herein lies the flaw. No physical problem can give 100% as an answer. Ever. There is a nonzero probability that you, sitting in that chair, will spontaneously explode before you finish reading this post. There's also a nonzero probability that you will quantum mechanically tunnel (pass straight through) through your chair, find yourself embedded in the center of the earth, and die a horrible death. There's a nonzero probability that the LHC will create dragons, and they will eat everyone. Should you be worried about these things? No. Why?
First, on the kinds of improbable events. There are more of them than real events. A lot more. In principle, I could count all the events that ever occur in the lifetime of the universe. It would be a big number. If I think of individual atomic interactions, there are about 1087 atoms in the universe. Let's assume each undergoes an interaction once per femtosecond (which is about what you expect for electronically bound materials like gasses, liquids, and solids on earth). The lifetime of the universe is about 1010 years, giving me 10119 particle interactions in the universe in its lifetime. Now, if I compute the probability that an atom will tunnel into the floor, and get 10-500, that means that on average, it will never happen during the lifetime of the universe. This, I think, is a reasonable definition of never given that I cannot get 0% or 100% as an answer.
Now how many events didn't happen? Infinitely many. Infinity is bigger than 10119. And this is the reason we cannot compute probability: we do not even know the set of all possible events. Consider if each of those 10119 events didn't happen, and add one more (say, dragons). The set of things that didn't happen is bigger than the set of those that did. The atoms of air in your room didn't undergo a nuclear reaction. You didn't spontaneously combust (though you're not to the end yet!), and dragons didn't eat you. How do I compute the probability of all these things that I've never seen happen? The best I could ever do is 1/(all the things that did happen). No amount of wishful thinking will get me a better answer, no matter how terrifying all those things that don't happen are.
Second, on small numbers. Do you have an intuition about how small 1% is? How about 0.01%? How about 10-500? How about 10-100000? Could you tell the difference between the last two? The last one is the kind of probability I'm talking about in the examples of the last paragraph. Let's make it a bit simpler, let's consider individual atoms, and the assertion that there's a nonzero probability for an atom to tunnel to the center of the earth. The probability of one atom at room temperature to tunnel through another is approximately e-(kinetic energy)/(potential barrier). For room temperatures of ~300 Kelvin and potential barriers of about the binding energy of electrons at 10 eV (I'm being generous), this gives probabilities of 10-10 or so. Now to get to the center of the earth the atom would have to do this millions of times, or (10-10)1000000 = 10-10000000. It'll never happen (using my above definition of "never").
So, back to the issue at hand. What's the probability that something never-before-seen happens today? It's incomputable. I could concievably assign the number 1/(everything that's happened), but given that the number of things that never happen is larger than the number of things that do happen, this tells me that the probability of something happening that never happened before is bigger than 100%! Clearly something is wrong. It doesn't make any sense to assign a probability to events that have never happened. I can't tell you the probability that the LHC will create a black hole, or that dragons will eat you. And, it's impossible to "guarantee" that improbable things won't happen. Should you be worried? No. If you still disagree, read my description of "never" over again.
This is the best we can do. We cannot ask statistics or science to give answers that it cannot give. We can however extrapolate from the set of things we've seen to place limits on the things that don't happen.
Science at its best, is a set of laws based upon the set of things we have observed to happen. The LHC black hole fears are predicated upon throwing out one or more of the laws we have devised from these observations. In particular thermodynamics, quantum mechanics, or time reversal. If I can create a black hole from two protons, why can't a black hole turn into two protons? The consequences of throwing out these pieces of physics are far greater than just black holes, and throwing out quantum and thermo are absolutely not justified by any experiment done so far.