Mathematically possible vs physically probable outcomes

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Related: https://math.stackexchange.com/questions/155156/is-it-generally-accepted-that-if-you-throw-a-dart-at-a-number-line-you-will-neve

Answer 4

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.)

Answer 5

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 $34\%$!

Answer 6

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.

Answer 7

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.