# Mathematically possible vs physically probable outcomes

A good buddy of mine and I have had a friendly debate about the origins of the current state of our universe (namely; Earth and life on Earth) and have fundamentally disagreed in our stances with respect to probabability, infinity in time and space and possible/probable event outcomes. He maintains the position that given a set of possibilities and enough trials, each outcome must have occured; which is his reasoning for why life must exist. I do not necessarily take issue with this particular concept, as given an infinite amount of time and states of matter life is bound to come from one of those states. In our discussions, however, we have been using a specific example in which I disagree vehemently with his stance. The example:

If you throw a handfull of sand in the air an infinite number of times, and that sand lands on a flat surface, every configuration will happen (according to his position). For instance, the sand landing in a pattern which spells your name out is a mathematically possible outcome, and will therefore happen given enough trials.

To counter his stance on this example, I took the position that there is a mathematical (but not physical) possibility that every grain of sand lands in the same one inch square of the surface; but I maintain that even though it is a mathematically possible outcome it will never happen because of the way the physical world works - that sand will be roughly evenly distributed for each throw, even if over an infinite number of trials, assuming consistent and fair trials (ie, no God or other being moving grains of sand). I submit that even though it is a mathematical possibility, you'll never see your name spelled out in block letter English anywhere in the universe without the influence of intelligence, even if you were able to attempt a verification for this - he disagrees. I held him liable for mathematical/physical proof of reasoning for his stance and he has taken to dismissing me as ignorant of probability and infinity. Can anyone provide some good reasoning for either side of this argument? I realize that either is an impossible stance to prove, since we can't verify our positions, but any well-reasoned insight will be appreciated. A similar question, with an answer I found to be relatively useful:

Infinite universe - Jumping to pointless conclusions

EDIT:

After reading some of the comments and answers here it has become apparent that I may have misrepresented my ultimate question. I realize that given a non-zero probability and an infinite number of trials, the mathematical probability of encountering the event described by said probability converges to 1. Some have taken the position that there is no disconnect between a mathematical probability and the likelihood (read: possibility) of a physical event happening.

To simplify the argument, the surface can be thought of as a grid - in which case every single configuration has some mathematical probability associated with it. My stance regards certain configurations as physically impossible, however, which is the reasoning behind my one-inch-square analogy. Can anyone show clear reasoning (and sources!) for their belief that it is possible to toss a handful of sand into a one inch square?

• I've deleted a comment discussion. To all involved: please keep in mind that comments are not the place for extended discussion; that sort of thing should happen in a chat room, e.g. Physics Chat. Jun 19, 2014 at 21:11

Physics uses mathematics. In a thought experiment of a machine ( to exclude the complications of hand throws that John mentions), using statistical mechanics any configuration is possible so even the telephone catalog on the floor with names and addresses. Note "thought experiment" .

Physics calculates the probability of this happening in very strict mathematical formulas and puts physical limits on what is probable to be seen using statistical measures, standard deviations, to gauge the confidence we can have on any measurement/observation in physics . We accepted that we saw the Higgs because the combined experiments gave five standard deviations for it not to be a fluke/coincidence. One chance in 100.000 . Experiments with such levels of confidence have not been falsified in elementary particle physics experiments ( from repeated statistically experiments, discovering errors is another story).

The probability of getting a name from the above thought experiment is effectively zero, for physics, if one puts down the number of permutations in space .

Thus the answer to your controversy is: you are both in a sense correct. Your friend is looking at the thought experiment, and you are looking whether it can be physically realized, and physics does put a limit to probabilities, experimentally, with what we know and have measured as probable in nature.

• This is the crux of my question. A mathematical probability does not ever turn into a physical certainty until it has happened and been observed. Making the assertion that somewhere in the universe is an exact representation, in block letter English, of my name is downright absurd - yet, the event is not impossible. Jun 16, 2014 at 8:15
• one enters metaphysics with these questions, or at best infinite probable universes, that cannot be tested. We have only tested 1 in a 100.000 in my example by we are sure that this is no coincidence, that two experiments get the higgs at the same invariant mass. Our sureness comes from the way we have experimented and tested the results against mathematical models. Physics excludes highly improbable events from physical realization. Jun 16, 2014 at 10:54
• It's quite obvious that the positions can't be tested. But, I'm looking for some solid reasoning for either stance; is it physically possible to toss a handful of sand onto a surface into a configuration which clearly, legibly displays a sentence in some language? I lean towards "no", because I feel that the sand will be roughly evenly distributed over any number of throws, as dictated by physical entropy. Jun 17, 2014 at 0:27
• We are excluding metaphysics, right? Then the only explanation for the DNA molecule, which is much more complicated than throwing a handful of sand and getting a word, is that small probability configurations do happen within our universe. And from DNA to go into a grown human, even smaller probabilities have to happen. Either one accepts that a lottery is a random thing, or one accepts a metaphysical world view, which essentially is "something" affecting/manipulating probabilities. Jun 17, 2014 at 3:09
• Physics puts limits at 1 in a hundred thousand or 1 in a million because physics is mainly about what can be experimented with and observed in the lab. There is a limit to the large volumes of mass, number of particles, and length of time that a human being can measure and experiment with, and this introduces limits to the probability of observable events. Jun 17, 2014 at 3:13

If the initial velocities of the sand grains are randomly distributed then there is indeed a non-zero probability that all the sand will land within a square inch of the surface.

I would guess that your objection is that the initial velocities are not randomly distributed. If I throw a handful of sand into the air then the shape of my hand, the way I throw, and probably many more parameters are placing constraints on the initial velocities. Those constraints will place limits on where the sand grains can fall, and it may well be that they exclude the possibility of all the grains falling in the same square inch.

The point is that if we have some configuration space for a system then there is a non-zero probability of find the system anywhere in that configuration space. However the probability of finding the system outside the configuration space is zero. The problem is that working out the limits of the configuration space is frequently hard.

• I know there is a non-zero mathematical probability, but this doesn't equate to a physical certainty over any number of trials. The tosses should be considered to be fair, as in the rest location of each grain of sand is truly, but naturally random. Another way to think about it might be to consider a volume of gas; my stance would be that under no normal circumstances would the particles of gas configure themselves to spell something clearly legible in some language. Jun 16, 2014 at 7:40
• Or maybe consider an electron when observed. I believe the electron will, under normal circumstances, never trace out the letters of the works of Shakespear from position to position. Jun 16, 2014 at 7:43
• It depends on what you mean by normal circumstances. Anything with a non-zero probability will occur if you wait long enough. Jun 16, 2014 at 8:29
• Well, I think that "long enough" has no meaning in physics when the time unit is an exponential of the time of creation of the observable universe. Jun 16, 2014 at 11:22
• I'm not entirely convinced that any non-zero probability will occur over $\aleph_{null}$ time. I realize it's philosophical, but imagine $\aleph_{null}$ universes. In one of them, it turns out that the sand grains always fall in a 3-meter diameter circle, every single time. Statistics are probilities, not guarantees. (PS that's one strange-looking "aleph." sorry :-) ) Jun 16, 2014 at 12:00

No, your friend is not right that "given infinite time, if there is a mathematical probability greater than zero, it will eventually happen". The probability converges towards 1, but there is no guarantuee that it will happen.

This reasoning is as wrong as saying that when i chose a real number X in [0,10] uniformly, then i will never end up with Pi, because P(X=Pi)=0 (since there are infinite reals in [0,10]). But that is true for any real, so... would i end up with no number at all? Of course not. As a sidenote, since i can make time countable if i descretize it, i could say that even when i sample out of [0,10] for an infinite amount of time (countably infinite samples), P(X=Pi) still is 0. It could still happen that i end up with Pi though!

The same is true in your example. I could infinitely throw sand and never get my name. Or i could get my name infinite times. Or i could get my name EVERY TIME. Each of these outcomes is possible (just not very likely). The law of big numbers only talks about probability, not about prophecy.

• Excellent answer, and this is what I was getting at in my comments to the OP. Jun 16, 2014 at 17:37
• I'd like if anyone could demonstrate that physical reality deals with real numbers (as you do here), rather than real intervals.
– user21433
Jun 16, 2014 at 19:24
• I read the OP's problem as being a version of the common conflation of "random" with "evenly distributed." I'm not sure measure-0 events are the problem, if the OP is willing to accept that slight variations on an extremely unlikely configuration still count as part of that configuration. There are many perfectly specified states of sand on a surface that I would recognize as "spelling out my name," and their combined measure is strictly greater than 0.
– user10851
Jun 16, 2014 at 19:35
• This is wrong. Assuming that you could randomly pick any number, there IS a 0% chance that it will be pi. What this tells us, is that in real life we can't pick a random real number. But back to the point. Given an event with probability > 0, it will happen after an infinite number of flips. This isn't meaningless conjecture that people throw around. If you can't wrap your head around this concept, then you don't understand infinity. The only analysis required in this question is whether or not any given arrangement of sand has p > 0 or not. That is up for debate, but your answer is wrong. Jun 16, 2014 at 20:59
• @Cruncher No, it will not necessarily happen (if you flip coins, the outcome could be HHHHHHHHH…), yes if you could pick a random number in [0,4] using the uniform distribution, the outcome could be π. You misunderstand what probability 0 means for non-discreet distributions. Jun 16, 2014 at 23:48

You and your friend don't seem to be on the same page regarding this discussion. When you say you wil never see you name spelled out in sand as the result of a random throw, you are not showing that the probability is zero, you are showing it is negligibly small in the frame of a human's lifespan (or indeed even the universe's).

This difference is important since, unlike what you seem to argue, there is no difference between mathematical probability and physical probability, given than you choose your mathematical model correctly. The physical world obeys mathematical laws one to one afterall. The only way the outcome of the throw is limited is by the nature of the randomness of the starting position. If the physical probability of an outcome is zero, that simply means that the mathematical chance of it happening was zero.

As a side note regarding your objection that it will never actually happen within the universe's life span: a multiverse theory could possibly solve that problem and have every mathematically possible outcome actually occur (given a normal distribution of the starting positions). (which would even include you performing the experiment, the sand quantum tunneling its way into your brain and altering your brain in such a way that you now DO believe that it is possible and suddenly develop a taste for peanut butter.)

In contrast to the other answers here, I'm not entirely convinced that the event will happen with probability one, even under the most idealized mathematical circumstances. These sort of infinite time probability situations can be really tricky.

For example, using the same reasoning one might conclude that a random walk returns to it's starting point with probability one if given enough time. That is true for 1D and 2D, but in 3D it only happens with probability $34\%$!

• Good answer, but i think the difference here is that in a random walk, the states are not independent. The next state depends on the state i just measured. In the experiment described by OP, one can at least assume independence (like a coin toss experiment). Jun 18, 2014 at 7:08

I think a crucial point here is that there is no infinity involved, at all.

The total mass of all matter in the universe is a rather big number, but it's not infinite. Gravity would not work the way it does, if that were the case.

The universe is very old, but it's not infinitely old (whatever that would mean).

So the question is moot, I think. Interesting, nevertheless.

• mass of all matter in the universe is a rather big number, but it's not infinite. You mean observable universe? Jun 18, 2014 at 7:35
• Good point. I would intuitively think the total mass of everything outside the observable universe is also finite, but there's no way to know, and the concept is probably meaningless, since its information (even its gravity) cannot reach is. Jun 18, 2014 at 12:27

In reality, everything in the universe has a non zero probability. There is a non zero probability that the I will vanish from the world in the next moment like I never existed. There is also a non-zero probability that I will find myself on the Martian surface in the next second. However, these are so small that they can be ignored. Thus, it is a pointless argument that can never be proven, as a truly infinite number of trials can never take place.

• Is this really true? I think that is what OP would like to get answered. Is it true that everything has nonzero probability, or are there constraints that completely rule out certain things? Jun 18, 2014 at 7:10
• In reality, everything in the universe has a non zero probability. So every physical law will be violated? Jun 18, 2014 at 7:34
• @jinawee Not everything. There are many things, which using the laws of physics can be proven to be impossible. Jun 18, 2014 at 16:03
• I was being rhetorical. Jun 18, 2014 at 17:35