Thursday, 4 September 2008

The Statistics of Improbable Things

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.


Hawkeye said...

I think I get what you're trying to say, but some of this is just plain wrong.

You say "nothing is 100% ever". I can guarantee you 100% that you will never find a circle of any diameter whose area exactly matches that of a square.

You say that it is incomputable to work out the probability of something that has never happened? I've never thrown a die and got the number 6 ten times in a row. But I can certainly calculate that probability. I going to stick my neck out and say that no-one has ever thrown a die one million times in a row and got the same number each time - but the probability of that can be calculated.

Maybe you wrote this in easy terms for the LHC doomsayers (or crackpots) to try and follow, but I don't think you've done yourself any favours by making such statements.

De Bunker said...

By "nothing is 100%, ever" I am referring to physical problems which can be computed in the context of quantum mechanics. Of course if I set up an idealized problem in which I can measure things with infinite precision, then your comment holds. But in the physical world, I certainly can find a circle whose area matches that of a square to the precision of my measurement instruments. (Your argument is that pi is irrational I assume, but I could never prove that pi is irrational by making a physical measurement)

With respect to rolling die, the reason these things are computable in your example is that you know you have a six-sided die (again you've created an idealized, non-physical situation). If you did not know how many sides the die had, you could not compute anything. (e.g. the set of possibilities is unknown, and even of unknown size) Furthermore, in a physical problem, a die can land on a side or corner or fall off the table or break in half, and these are perfectly acceptable physical answers not contained in your assumed state space. Simply rolling die and seeing 1-6 does not give me the capability to compute the probability that the die will crack in half.

So I stand by my statements (and I have revised my post slightly to clarify, I hope). And yes, I'm trying to be as non-technical as possible.

De Bunker said...

For the experts: This is Bayesian vs. Classical statistics. When evaluating the unknown, we must use Bayesian. We don't even know the system, or the set of possible outcomes, and therefore cannot make Classicist predictions about unobserved events.

But we can make Bayesian predictions. This is the origin of the 1/(everything that's happened) argument of my post. This is a Bayesian prediction.

Erik Martin said...

"I can guarantee you 100% that you will never find a circle of any diameter whose area exactly matches that of a square."

Huh? What about a circle with a diameter = 2, and a square with a length = the square root of pi?

"I going to stick my neck out and say that no-one has ever thrown a die one million times in a row and got the same number each time - but the probability of that can be calculated."

Yes, but an analogy comparable to the the LHC disaster would be rolling a 6 on your first million rolls, and then on your million and first roll, rolling the square root of 2. Calculate them odds.

Alejandro Rivero said...

"What's the probability that something never-before-seen happens today? "

Actually, it is 100%. On the contrary, the possibility of something that happened yesterday to happen today is zero.

And more seriously, your argument about the finite number of atoms in the universe &c could be refined to argue that there are some measures with exact 100% probability. It is not more rare that having exact 50% or exact 25%. And surely you can get some of this kind of exactness when topological arguments, and specially index theorems, are put to play.

Thomas D said...

Do you know who at CERN is responsible for marketing the LHC as a 'big bang machine' or saying it will 'recreate conditions in the early universe'?

Those are two pretty damn inaccurate statements and they make it a lot harder to defend CERN safety assessments, since if CERN can exaggerate in this respect why should we believe it is not exaggerating in others?

De Bunker said...

Alejandro, like "hawkeye" you are creating an artificial (and unphysical) system in which you can know the entire state space (regarding topological whatnot). In the absence of an absolutely correct theory, we must content ourself with Bayesian predictions based on our collective observations, and at best this can make predictions with no more than the precision to which we've measured. (e.g. you can write down almost anything at all with a femtobarn cross section)

And thomas, these statements about the "big bang machine" make me a little uncomfortable too, but are technically correct. Our understanding of gravity indicates that at the energies we are reaching, the universe was in this state at the earliest times. Of course for physicists, the fact that the big bang occured and had those temperatures isn't as interesting as what particles are participating in the dynamics up there.

Actually I think this term is being used by the BBC, not by CERN's press office. But, correct me if I'm wrong...

Thomas D said...

The BBC hopefully wouldn't just make up something like that without some guidance or approval from CERN... besides, quotes linking LHC to 'big-bang recreation' have been around for some months or years.

I don't think it is technically accurate either:

Conditions after the Big Bang were homogeneous and thermal in a slowly (by particle physics standards) expanding geometry. Proton collisions, which are the 'discovery' mode, can never be thermal.
Whether the heavy-ion stuff gets at all close to equilibrium remains to be seen, but these collisions expand under their own steam much faster than the early universe plasma did, and are probably not very homogeneous.

The chemical potential is wrong too, the early universe was almost exactly symmetric in matter/ antimatter.

Best we can say is that both the Big Bang plasma and a heavy ion collision are, for a short time, highly energetic and full of quarks and gluons.

Anonymous said...

Isn't there a zero probability that it will form a black hole at a precisely measurable point in spacetime?


Pawl said...

I hope we can use this blog not only to debunk some of the end-of-the-world ideas but also to examine more carefully some of the arguments which have come up in the course of this.

I'd like to sort out whether this claim that Hawking radiation is a consequence simply of CPT invariance (or of T invariance) is valid. The version I've seen seems to be in error.

The argument made is that if there is an amplitude

particles -> black hole

then there is an amplitude of the same magnitude

black hole -> (anti-) particles.

The problem with this is that the time-reverse of a black hole is not another black hole but a white hole. So what CPT invariance really says is that there is an amplitude of the same magnitude

white hole -> (anti-) particles.

That is true but hardly a surprise. (And it says nothing about stability of black holes.) Can anyone improve on this?

Of course, this whole argument only arises to the extent CPT invariance holds. While it is certainly true that CPT invariance is both experimentally well verified and theoretically justified in special relativistic qft, in the context of micro black holes we cannot avoid thinking about quantum gravitational effects. No one knows whether CPT will be respected by quantum gravity. Violations would not be any more in conflict with either theory or experiment than was the discovery of P violation in the weak interactions. said...

What is it about people who clearly know what they know, and use defendable methods to arrive at reasonable guesses, and occasionally, proofs, that seems to bring out the worst in people who do neither and in huge disproportions will happily defend psuedo-science (will list their astrological sign on their blog). Perhaps there is a "Star Trek" future for humanity when these logical lesions will succumb to their own suffociation, but it's unlikely to happen as long as those who are capable of understanding the present continue to work hard to prepare a safe future for all.

Vincent said...

When you say that in the reality, a probability strictly equal to 0 or 1 doesn't exist, people really don't like that, I don't truly understand why, but for sure, they want absolute certainty.

Baragon-Kun said...

Well its that true that people usually do not understand that this is the most closest thing to say Zero, i mean, i dosnt matter if there is a 1 in the statics, depends on how much opposite numbers are in the other side

1 against 1 000 000
1 against 10 000 000
1 against 50 000 000

what does this tell you???

but yeah, people usually think that if there is a "1", may "can happen".

RELAX people, this people know their shit, we must trust on them

Bob said...

Doesn't matter about statistics, you can't know what will happen once the LHC collides atoms. Maybe it will be nothing. Maybe we'll discover some awesome particle to further science. Maybe we'll discover an exotic particle that will rip a whole in space-time continuum that will destroy the planet.

The fact is, what do we really know about colliding atoms? What do we really know about the universe.

From the many lectures and documentaries I watched, we apparently know NOTHING. You may think we know something, but we can't even create an efficient way to create fusion energy, and we're here colliding atoms like as if we are smart or something.

Shouldn't we first figure out how to land people on Mars before we start colliding atoms to figure out the origins of the universe?