Infinity and absolute probability

**chbrules** · July 16th 2008, 12:52 AM

A friend and myself are arguing about the probability of the least probable (1/infinity) outcome given infinite chances.

The whole argument revolves around the premise the universe is finite. Because it is finite, occurences of events with low probabilities (such as 1/infinity) may never occur. However, if the universe was infinite then everything would always occur.

I'm not sure how to mathematically represent this so here is what I've done simply:

P(x) = 1/infinity
t = infinity

P*t = 1

Is this true? Could someone explain this if I am wrong? Thanks!

**carolus4** · July 16th 2008, 03:22 AM

I have thought along the same lines but I have been shown only frustrating non-results.

So that you may share my frustration inf * 1/(inf) is undefined:

consider limits as n tends to infinity:
if you have n one one side and 1/n² on the other, you consider the limit of their product which is simply the limit of 1/n ie 0 (or 1/infinity)

if you had n² on top and n on the bottom you would fnd the result to be infinity

if you had a*n on top and n on the bottom you would get a

Frustratingly, all of these examples correspond to inf/inf and have different answers.

To find practical solution you need to have a non infinite comparison between the two, usually just find an n and consider the limit of the product.

The interesting thing I did gain from this which is not fully relayed is the fact that some infinite spaces are larger than others, which is a handfull to visualise.

**meymathis** · July 16th 2008, 06:37 AM

Well one thing is that $\infty$ is not a number. Infinity is a concept, not a number. For one thing there are different levels of infinity as carolus4 noted. For example, the rational numbers and the real numbers are two sets with infinite size but the reals are much "larger" than the integers. You can use infinities as the bounds of sums and integrals and in limits, but they really only tell you that you are going off unbounded. When we do limits like

$\lim_{n\rightarrow \infty} \frac{1}{n}$

sometimes you will hear people say this is $\frac{1}{\infty}=0$ . This is not really right, although the idea is basically right, and it certainly give you the right answer. Really, we mean than as n gets big, $\frac{1}{n}$ gets infinitely small. Or you can think of it as a limit of the form $\frac{1}{\infty}$ which are 0.

This may sound like semantics but its a little more than that. For example,
$\lim_{n\rightarrow \infty} n^2-n=\lim_{n\rightarrow \infty} n^2-\lim_{n\rightarrow \infty}n = \infty - \infty =0$
IS WRONG. This just goes to show that infinity cannot be treated like a number.

Second, if I understand what your question is, you are asking whether events with probability 0 can happen. And the answer, surprisingly is YES. Even in a "finite" space. By finite, here I mean bounded. However, you do need infinities running around to make it work. What you need is an uncountablely infinite (like the reals or irrationals) set of outcomes, and not for the outcomes to be bounded. So if space or time is real-valued, than you could have things occurring with 0 probability.

As a "realistic" concrete example suppose we have a particle sitting around waiting to decay, and the particle is moving in a circle. The time of decay is a exponentially distributed, and the location of the particle when it decays (given that the decay time is sufficiently long compared to time it takes to move around the circle) is uniformly distributed. The random variable that we are interested in is the place on the circle where the particle decays. Ok, now we wait and after 5.39874627624287... minutes it decays at some point, x in space. It turns out the probability that the particle decaying in that spot is 0. If C is the circumference of the circle, the probability is given by

$\lim_{\epsilon\rightarrow 0} \int_{x-\epsilon}^{x+\epsilon}\frac{1}{C}dy = \lim_{\epsilon\rightarrow 0}\frac{1}{C}(x+\epsilon-(x-\epsilon) = \lim_{\epsilon\rightarrow 0}\frac{1}{C}2\epsilon = 0$

In other words, we just witnessed an event with probability 0!!!!!

Now what is true is that we saw an event where the probability density function is positive, not 0. But the probability of the event is definitely 0. Because we have an uncountable set of outcomes, this is the only way for that to happen. If each spot on the circle had a positive probability, there is now way for the total probability to be 1. See http://en.wikipedia.org/wiki/Series_(mathematics)#Real_sequences

Now where your intuitions I think are hitting closer to the mark is this: lets say you ran that experiment once and you record the exact location and now you are going to run the experiment over and over again until it happens to decay at the exact same spot (or any other spot that you choose before hand). You WILL wait forever, because the probability is 0. (number of times you need to repeat the experiment has Negative binomial distribution with p=0).

Now even though I called this concrete, it really is theoretical. We certainly can't measure the time and or location of the particle to an absolute level of accuracy. In fact, quantum mechanics tells you that you CAN'T! Second, we don't understand space well enough to even say whether it is real or rational, or something else. So take heart. If time and space are rational, then those 0 probability events can't happen.

**ThePerfectHacker** · July 16th 2008, 08:13 AM

I am no probability expert - very far from it. Here is an attempt to answer the question. It all comes down to how mathematicians define probability. In high school (or college) when you probability, you thought of probability as the number of favorable outcomes divided by the number of possibile outcomes. For example, if you want to get a sum of 3 on a pair of die you need to do two steps. First, let us give each die a name, calle one die #1 and die #2. Second, the number of possible outcomes if different outcomes with the die #1 and die #2 which is 6*6=36. Third, the number of favorable outcomes only happens if die #1 is 1 and die #2 is 2 OR die #1 is 2 and die #2 is 1, thus there are only two favorable outcomes. Fourth, compute 2/36 = 1/18 and that is your answer. This is the simple type of probability. Let us look at a different probability problem. Say we have a coordinate square: $0\leq x\leq 2$ and $0\leq y\leq 2$ . And we throw a dart randomly and wish to hit: $0\leq x\leq 1$ and $0\leq y\leq 1$ i.e. the bottom left quarter. The answer is going to be 1/4. But how do we get that answer by counting the # of favorable outomces? There are infinitely many of them! But eventhough there are infinitely many possible outcomes - and thereby the old definition for finite probability does not apply - mathematicians have a way of still finding the probability. I think they are called "Measures" and "Lebesgue integration", let us not worry about these are, just think of them are approaches of summing over infinitely many points. Mathematicians define the most general version of probability (even for infinites) by the approaches above. That is how mathematicians decided to define it. And since we picked such a definition we will have some consequences which you might find shocking. Consider the coordinate rectangle again, now ask thyself of what is the probability of hitting the line $x=1$ ? It turns out the the probability is 0. But I know what you are going to ask: "it is still possible to hit the line?!?!". Before I gave an answer to that question here is another question. Suppose you have the coordinate rectangle again. And you want to probabity of hitting a point $(x,y)$ where both $(x,y)$ are rational numbers. It will turn out the probability is zero! Again you will ask the same question. To give an answer to this question, again it all comes down on what definition we choose for probability. Using the mathematical definition of probability we get 0. It cannot be wrong because it is just a definition, but still whenever we introduce a definition we try to make it consistent with how we thing the world should work. And you might argue that the answers we got are not consistent. But is this really true? Remember we are throwing a dart which has an infinitely small point, and we are trying to hit a line which has no thickness to it at all, or we are trying to hit a perfect rational point i.e. a dot with no size to it. Are these things you actually find in the world? No, there are no such things. The two problems that I posted with probabilities 0 are just in the imagination they do not even exist in reality. Here is another example of a strange answer that I came up with a long time ago: say you pick any positive number (integer) what would be the probability that I will guess it correctly (it is 0 again, eventhough it is possible that it is the same number).

**meymathis** · July 16th 2008, 08:27 AM

Although it pretty much doesn't matter how you define probability, I think. If you want to have the basic rules of probability work, it implies certain things. If you have an uncountable number of outcomes, the number of outcomes with positive probability would have to be at most countably infinite.

Also, I'm curious what your probability distribution looks like for your last example. I don't think I've ever seen a countable outcome space for a random variable that had probability 0 everywhere. Not that I'm saying that it is wrong, I just have never seen it.

**CaptainBlack** · July 16th 2008, 08:30 AM

Originally Posted by meymathis

Second, if I understand what your question is, you are asking whether events with probability 0 can happen. And the answer, surprisingly is YES. Even in a "finite" space. By finite, here I mean bounded. However, you do need infinities running around to make it work. What you need is an uncountablely infinite (like the reals or irrationals) set of outcomes, and not for the outcomes to be bounded. So if space or time is real-valued, than you could have things occurring with 0 probability.

What makes you think that you need uncountable sets here, what is wrong with a countably infinite sample space?

RonL

**CaptainBlack** · July 16th 2008, 08:34 AM

Originally Posted by meymathis

As a "realistic" concrete example suppose we have a particle sitting around waiting to decay, and the particle is moving in a circle. The time of decay is a exponentially distributed, and the location of the particle when it decays (given that the decay time is sufficiently long compared to time it takes to move around the circle) is uniformly distributed. The random variable that we are interested in is the place on the circle where the particle decays. Ok, now we wait and after 5.39874627624287... minutes it decays at some point, x in space. It turns out the probability that the particle decaying in that spot is 0. If C is the circumference of the circle, the probability is given by

$\lim_{\epsilon\rightarrow 0} \int_{x-\epsilon}^{x+\epsilon}\frac{1}{C}dy = \lim_{\epsilon\rightarrow 0}\frac{1}{C}(x+\epsilon-(x-\epsilon) = \lim_{\epsilon\rightarrow 0}\frac{1}{C}2\epsilon = 0$

In other words, we just witnessed an event with probability 0!!!!!

You cannot measure an arbitarily precise time and position, never mind quantum uncertainty, instruments do not work that way, so this not realistic or concrete, but is still a thought experiment - best go away and think about it some more.

RonL

**CaptainBlack** · July 16th 2008, 08:46 AM

Originally Posted by chbrules

A friend and myself are arguing about the probability of the least probable (1/infinity) outcome given infinite chances.

The whole argument revolves around the premise the universe is finite. Because it is finite, occurences of events with low probabilities (such as 1/infinity) may never occur. However, if the universe was infinite then everything would always occur.

I'm not sure how to mathematically represent this so here is what I've done simply:

P(x) = 1/infinity
t = infinity

P*t = 1

Is this true? Could someone explain this if I am wrong? Thanks!

Probability is a mathematical theory which is used to model some aspects of reality, and is useful to the extent that it does model reality. But when the application of a tool like probabilty to a thought experiment leads to an absurdity there are a number of possibilities - the thought experiment is nonsense - the tool is inappropriate - ...

RonL

**chbrules** · July 16th 2008, 12:21 PM

Thank you for these wonderful replies, you guys!

Perhaps I need explain our disagreement a bit further. This is actually a philisophical argument, but I was interested whether or not that we could represent it within the bounds of our mathematics.

In tautology we have an understanding that we work in the relative universe where we don't have absolute knowledge. Within that scope we understand that we cannot actually disprove anything because we could never measure all possibile outcomes of anything. Thusly, something is always true, regardless of how minute it may be in probability. This is where I tried to derive the least probable outcome 1:Infinity.

Then, I proposed that the universe's length of time would have to be finite, else given an infinite set of chances that the least likely probable outcome would become 100% probable over unbounded length of time.

Originally I heard this argument on a NOVA series from a quantum mechanics professor at some university. He was the host of the 3-part series, The Elegant Universe. He proposed that if you push on a wall for an infinite amount of time that you would eventually go through it, but, since we don't have infinite time, our chances of going through are virtually 0.

**ThePerfectHacker** · July 16th 2008, 12:38 PM

Originally Posted by chbrules

This is actually a philisophical argument, but I was interested whether or not that we could represent it within the bounds of our mathematics.

That is a problem. Philosophy has nothing to do with mathematics.

Originally I heard this argument on a NOVA series from a quantum mechanics professor at some university. He was the host of the 3-part series, The Elegant Universe. He proposed that if you push on a wall for an infinite amount of time that you would eventually go through it, but, since we don't have infinite time, our chances of going through are virtually 0.

I seen that too. I have no idea what Brian Greene meant by that. A problem with "Popular Mathematics" and "Popular Physics" is that the authors try to explain things in the easiest possible way to understand because it is written for people who have no experience in mathematics/physics. As a result, sometimes they say things which do not really make too much sense because they have difficultly in trying to explain them elementary. I am guessing that Greene was in a similar situation, he wanted to explain probability in quantum theory, and was no sure how to do with it.

**icemanfan** · July 16th 2008, 12:45 PM

Here is my answer to your question. There is no such thing as a probability of $\frac{1}{\infty}$ . You can have probabilities that are as small as you like. In the case that the probability of something occurring is always greater than zero, then it will happen given enough chances. But if the probability of something happening is always 0, it will never happen no matter how many chances you give it, even if that quantity is unbounded in time. If the probability of an event changes in time, it may or may not happen, depending on how many chances it gets.

**CaptainBlack** · July 16th 2008, 01:19 PM

Originally Posted by icemanfan

Here is my answer to your question. There is no such thing as a probability of $\frac{1}{\infty}$ . You can have probabilities that are as small as you like. In the case that the probability of something occurring is always greater than zero, then it will happen given enough chances. But if the probability of something happening is always 0, it will never happen no matter how many chances you give it, even if that quantity is unbounded in time. If the probability of an event changes in time, it may or may not happen, depending on how many chances it gets.

But you are wrong, for instance the uniform distribution on the interval $[0,1]$ gives non-zero peobability for an event corresponding to finding a value in an interval $[a,b]$ where either $a \in [0,1]$ or $b \in [0,1].$ .

But now the probability of a particular value $c$ in $[0,1]$ occuring is zero.

Now this may not be physically significant as any measurement has an uncertainty associated with it so effectivly samples a neightbourhood of some point, but the mathematical theory does not care.

Thread close, as it has degenerated into a pointless philosophical discussion.

RonL

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