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December 5th, 2015, 02:01 AM   #1
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Wondering about probability in an infinite set

Looking at an old online discussion about picking a random item from an infinite set, I had a thought about this so I thought I'd post it somewhere to see what other people think. I accept that I might not know too much about this, and this could just look like nonsense.

The basic question I am thinking about is, if you pick a real number at random from a particular range, what is the probability of it being some pre-determined real number? I was trying to decide if the probability is zero, or if you can't pick a random number from an infinite set at all.

I've decided you can pick a number at random, but the probability of them being equal is zero. I figure the issue is specificity. Here is my reasoning.

Consider the range [0,1]. I think it may be possible to pick a real number at random from this range. Start by dividing into 10 equally sized ranges: [0,.1],(.1,.2],(.2,.3],...(.9,1]. We can assign the random real number to a particular range by choosing a first decimal digit (after the decimal place) at random. Then we can narrow down the range further by choosing the next decimal digit at random. In this way, no matter how finely we divide up the initial range of [0,1] into smaller ranges, the probability of our random number falling into any one of them is always given by the size of the range. So if we repeat this an infinite number of times, it seems we've generated a real number at random.

If you have a pre-specified number in that range, and a way to check equality against this number, I think you'll find the probability of choosing it is zero. I don't know if this means it can't ever be chosen - I'd assume this is the case.

In fact, because the progression of digits is chosen at random, there is no way to specify the randomly generated number completely. It is not only transcendental, but lacks specificity. It can only be placed within a smaller and smaller range of real numbers.

What do you think?

The next thing I am wondering about is if it makes sense to choose numbers at random from a discrete infinite set, like the positive integers.
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December 5th, 2015, 05:48 AM   #2
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This is one of the big problems that probability has. The probability of picking a particular cannot be zero for all numbers in the interval, because that would mean that it isn't possible to pick any number at all. But clearly, if there is an infinite set of numbers, each having the same finite non-zero probability of being chosen, the cumulative probability of selecting from that set is greater than 1. So that's not possible either.

Things get a little more complicated for non-uniform distributions, but the flavour of the problem is similar

My take on it would be that in real life we never choose from truly infinite ranges. If you throw a dart at a dartboard, there is a limit to the accuracy with which we can measure its position. And the point covers an area anyway. If you select a number between zero and one, it will come from a finite domain in practice. You certainly won't pick most numbers, because representing them would require more particles than there are in the observable universe.

In the distinct case we get a similar problem. In tossing a coin, we say that in theory there is no limit to the number of "heads" you can get in succession. So we give a finite non-zero probability for each number of successive heads. But in practice you can't get most of these because of issues like the longevity of you, your coin, the universe, etc..

Thus infinite distributions provide a convenient approximation.
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December 5th, 2015, 12:40 PM   #3
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Fill a 1-cup measuring cup with marbles. You can count how many marbles are at the 1/2 cup line. Take the marbles out, and fill it with water. What volume of water is at exactly the 1/2 cup line? Can't answer? Well, it's the same issue.

And it is not a problem at all - it is the difference between discrete measurement (counting marbles) and continuous measurement (volume of water). Nor is it an approximation. In your question, it is the difference between a discrete probability probability problem, and a continuous one. Discrete distributions assign probability to each specific value. Continuous ones do not, they assign probability only to a range of values. Just like you can only measure the volume between two lines on the cup.

The paradox that you think you see is a different issue, requiring limits. It really isn't the topic for this forum, so I won't go into it in more depth. The point is that any range of values has an infinite number of real numbers in it. If a non-zero probability were assigned to each, the probability for the entire range would be infinite. Which is absurd.

Instead, probability is only assigned to a range. If a wheel-of-fortune is spun, with possible results measured in degrees around the wheel, the chance the result is between A and B (where B>A) is (B-A)/360. While it is true that sqrt(15,743)~=125.4711 has zero probability, it is also impossible to say that you have achieved that exact value. The best you can do is say the value is in the range (sqrt(15,743)-x/2,sqrt(15,743)+x/2), where x is a very small number. This range has a probability x/360.

Last edited by JeffJo; December 5th, 2015 at 12:45 PM.
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December 6th, 2015, 07:37 AM   #4
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If xy = 10, both x and y can get infinitely close to 0 but will never be 0. I don't know if it's mathematically correct to say this, but I would say that the probability of a randomly selected real number being 0 is infinitely close to 0 but will never be 0.
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December 6th, 2015, 02:27 PM   #5
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No. In a continuous probability distribution, we can talk about the probability that x is in a given interval but the probability that x is a specific number is always 0.
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December 7th, 2015, 03:42 AM   #6
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Let me try my explanation a different way. (I had left one concept intentionally ambiguous, fishing for a question that would indicate the reader's level of understanding.)

You have a cylindrical flask of radius R, and height H. It is filled with water. What is the volume of water between lines drawn on it at a*H and b*H, where 0<=a<=b<=1?

This is simple geometry: pi*R^2*(b-a)*H. In particular, if a=b it is 0. That doesn't mean there is no water at that level; after all, the flask is full of water. It only means that the measure of the water's volume is zero.

In a continuous probability distribution, we assign values called a probability density function, f(x), to each point in the continuous range. It measures the contribution to probability at any point x, not the actual probability. Just like the contribution to volume at level b is the area pi*R^2. The probability is like the volume in my analogy; it is only detectable over a range. A "volume of probability" of zero does not have to mean "no probability," just that the range we choose can't detect any because it has no length over the continuum we use.

Does that help?

Last edited by JeffJo; December 7th, 2015 at 03:45 AM.
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December 7th, 2015, 10:23 AM   #7
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Hi Jeff,

Quote:
I had left one concept intentionally ambiguous, fishing for a question that would indicate the reader's level of understanding
I am not in high school - I might have posted this in the wrong place. But thanks for the analogy. I wrote a paper while in university about some of the history of the description of real numbers, and I like to think sometimes about this issue of specificity.

Random real number generation seems like an interesting thing to think about. I'm guessing a random real number generator like I described in the first post could not generate specifiable numbers (ie rationals, etc) because, being random, there has to be a full measure of entropy in the sequence of decimal digits it generates, and in any specifiable number this is violated. I'm not sure if that is the right terminology.

But if that's the case, there are a whole collection of real numbers in the given interval [0,1] that could never be chosen by the random number generator. So it seems like the generator doesn't 'cover' the whole interval, so maybe it is invalid?

Last edited by mdg583; December 7th, 2015 at 10:29 AM.
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December 8th, 2015, 04:44 AM   #8
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Quote:
Originally Posted by mdg583 View Post
Random real number generation seems like an interesting thing to think about. I'm guessing a random real number generator like I described in the first post could not generate specifiable numbers (ie rationals, etc) because, being random, there has to be a full measure of entropy in the sequence of decimal digits it generates, and in any specifiable number this is violated. I'm not sure if that is the right terminology.
If you want to actually achieve any real number in [0,1), then you need my wheel-of-fortune, and you can only represent it as the analog ratio of the wheel's position to 2*pi. (I'm using radians to emphasize that the result is not limited to being rational). Heisenberg's Uncertainty Principle aside, you can't measure the numbers to get a decimal (or binary) representation of it.

If you use a computer, you have a discrete distribution because it can only produce numbers that are a ratio to 2^n. The problems you originally described apply only to the analog version I described, and I was addressing it as though that is what you meant. The point is that the "probability of an exact value" is a concept that only applies to discrete cases. It's a subtle point that often produces this exact question. I even had this same conversation with my son, now in med school, after his first course in probability.
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December 8th, 2015, 05:42 AM   #9
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Tldr;
You can't generate a random real or even a random rational or a random natural number, be aide doing so is an infinite process.

Leaving aside Heisenberg, and leaving aside the open question as to whether space is discrete or continuous, your wheel is still not capable of generating a random real number (under any distribution). We can't even generate a random rational between 0 and 1. This is because you can only measure to a finite precision. It's natural to consider measurements under the decimal system in discussing this, so that is what I will do initially. But the same is true for any measurement system, of which more later.

The problem comes because we can't measure an infinite number of decimal places. Measurement is a discrete process where we refine our measurement a finite number of decimal places at a time. You will never know if your wheel stopped at $\mathrm e$ radians. If you measure it accurate to 100 decimal places, you still don't know if the actual position of the wheel differs from $\mathrm e$ in the thousandth decimal place or the millionth. And since there are uncountably many reals that do match the 100 decimal places you measured, you have just guaranteed that those numbers almost all of them will never appear from your generator.

It doesn't matter whether you measure in decimals or binary, or divisions of powers of $\mathrm e$. You are limited by the accuracy of your measurement - and you will, by definition, never return a measurement if to try for infinite accuracy. You can even use a finite combination of systems, but the same problem reoccurs. Infinite accuracy takes infinite time.

Initially, it's tempting to suggest that we might use a discrete random variable with a distribution over the positive integers to generate a random rational between zero and one.

E.g. take two numbers from a Poisson process such as the number of radioactive emissions from a lump of radium in a fixed time. But these are only approximately Poisson distributed. There are only a finite number of decays that can happen from a finite amount of material. And even if there weren't, there is certainly a limit to the number of emissions we can count in a finite time.

A second example would be taking (one more than) the number of consecutive heads from tossing a coin. But gaian, this is not a true geometric distribution. The sequence HHHHH... Is possible, but we will never know if we have it. In order to return a value we must at some point stop. And at that point we lose the infinite distribution.
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December 8th, 2015, 08:10 AM   #10
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Quote:
Originally Posted by v8archie View Post
...your wheel is still not capable of generating a random real number (under any distribution). We can't even generate a random rational between 0 and 1. This is because you can only measure to a finite precision.
I'm pretty sure I said as much (btw, I did so in an attempt to preempt pedantic arguments like this - oh, well). Still, I'm pretty sure that wheel stopped somewhere, and that that somewhere represents a random real number in [0,1).

I didn't say you could measure it. Again, I'm pretty sure I said you could only place it in a range.

Quote:
It's natural to consider measurements under the decimal system in discussing this, so that is what I will do initially.
Maybe so, but that is the approximation. Which is why your argument is irrelevant. In the Mathematics of Probability (as opposed to Metrology), continuous distributions do exist, and are quite useful.
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