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unreality

Question

I have a few problems to scratch your heads over :P

1) You pick a random number from an infinity of integers, ie: 1,2,3,4,5...etc. What's the chance of picking 11,734?

2) You pick a random number from 0 and infinite integers, ie: 0,1,2,3,4,5...etc. What chance do you have of picking 11,734?

3) You pick a random number between 0 and 1 (including 0 and 1). What chances do you have of picking 0? of picking 1? of picking a whole number in general (0 OR 1)?

4) You have a random number generator that generates a random digit, ie 0-9, with a 1/10 chance per digit. You run the generator to infinity, creating an infinite stream of random digits. What is the probability that there are absolutely no 3's in the digit stream?

have fun :D

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Interesting statement. Let's pursue it a bit.

The set of reals in the interval [0, 1] is infinite.

I choose the number 0.314.

Prove that it was not chosen at random.

Tell me how you chose it, and I will tell you whether it is a random number chosen from all members of [0,1]. :P

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Tell me how you chose it, and I will tell you whether it is a random number chosen from all members of [0,1]. :P

I chose it at random.

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I chose it at random.

There are several ways of selecting a random number on the interval [0,1]. If your method is different from one of the below, let me know.

1. Banging away at the number-pad on your keyboard. If this is the method you used, then I contend you were selecting a (quite possibly random) number from only a subset of the real numbers on [0,1]. I say this because (I assume) you are only capable of typing a finite number of digits in a finite measure of time, and are therefore incapable of selecting the majority of real numbers (irrational numbers, for example). If the example you give means that you only had time to generate three numbers, then you can only say that you randomly selected a number from the set of real numbers on [0,1] that have all zeros after the third digit.

2. Throwing a dart. Your ability to determine the point on which your "dart" landed depends on the precision of your measurement instrument. This is necessarily finite, unless you believe in the continuum hypothesis (the one from physics, not mathematics). You can therefore only give a finite range of values in which your dart landed. If the example you give represents the precision of your measurement, then all you can say is that you selected a number between 0.3135 and 0.3145.

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I think the problem here is that most people don't understand that 1/infinity, taken by itself, is a meaningless statement, and we really can't say it's equal to ANYTHING. The statement "1/infinity" is shorthand, not actually trying to say, "1 divided by infinity." Infinity is a concept, not a number. For proof of this I look to http://mathforum.org/library/drmath/view/62486.html

"Q: If I take a chocolate bar and divide it among an infinite number of people, how much does each person get?

A: What if you divide it among every person living on earth? Each person would get roughly 1 six-billionth of a chocolate bar. That's a very, very small amount, and you'd probably need a microscope to see your piece, but it wouldn't be zero, right? Ah, but you asked about dividing it up amongst an infinite number of people. Well, we can't. Why? Because infinity isn't a number, so you can't show me an infinite number of people. If you try to, I will just add one more person, and then we'd realize that the number you thought was "infinity" actually wasn't."

I have to say, I was not clear in my original answer. The answer is "You can't do that," Rather than, "you have 0 probability of doing that," which are technically different answers.

You can't randomly choose a number from an infinite set. (I might argue that you can't really choose a random number at all, which is the subject for a different brain teaser.) People can't do it because we're pretty biased against numbers like 1024885926414553033465570345667158541759501107168465168464013834058687365 simply because they take too long to say/write and we have things to do rather than continue typing nonrandom digits, and obviously once we've established that some numbers are more likely to be chosen than others we know they're not random.

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Wow...why do the probability threads always have the most heated arguments? What is the probability density function for how much arguments go on as a function of how much probability is involved in the question?

;P

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There are several ways of selecting a random number on the interval [0,1]. If your method is different from one of the below, let me know.

1. Banging away at the number-pad on your keyboard. If this is the method you used, then I contend you were selecting a (quite possibly random) number from only a subset of the real numbers on [0,1]. I say this because (I assume) you are only capable of typing a finite number of digits in a finite measure of time, and are therefore incapable of selecting the majority of real numbers (irrational numbers, for example). If the example you give means that you only had time to generate three numbers, then you can only say that you randomly selected a number from the set of real numbers on [0,1] that have all zeros after the third digit.

2. Throwing a dart. Your ability to determine the point on which your "dart" landed depends on the precision of your measurement instrument. This is necessarily finite, unless you believe in the continuum hypothesis (the one from physics, not mathematics). You can therefore only give a finite range of values in which your dart landed. If the example you give represents the precision of your measurement, then all you can say is that you selected a number between 0.3135 and 0.3145.

You're pulling my leg now, right?

I mean, how can there be several ways to do what is impossible?

First I chose the number of decimal places. Out of the positive integers, and at random, I chose 3.

Then I chose a three-digit number from among the 1000 choices. At random, I chose .314.

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You're pulling my leg now, right?

I mean, how can there be several ways to do what is impossible?

First I chose the number of decimal places. Out of the positive integers, and at random, I chose 3.

Then I chose a three-digit number from among the 1000 choices. At random, I chose .314.

Not that I'm arguing with you about the probabilities part, I agree with you on the theory...but just can't noting how 3.14 are the first digits of pi...so I can't help thinking maybe your choice in this case was affected by your subconcious...;)

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You can't randomly choose a number from an infinite set.

(I might argue that you can't really choose a random number at all, which is the subject for a different brain teaser.)

People can't do it because we're pretty biased against numbers like 1024885926414553033465570345667158541759501107168465168464013834058687365

simply because they take too long to say/write and we have things to do rather than continue typing nonrandom digits,

and obviously once we've established that some numbers are more likely to be chosen than others we know they're not random.

Hi S_W,

From my other posts, you may be aware that I'm interested in proofs

that you can't choose a number at random from an infinite set.

You seem to offer such a proof in your post, and I'd like to explore it.

If I understand your post, you prove your main assertion by making a secondary assertion.

Namely that, among the set of positive integers, elements have differing likelihood of being chosen.

Let me choose one at random: 314.

If I understand your post, you would object to 314 as a random choice,

because you have established that the probability of choosing 314 differs

from the probability of choosing say

1024885926414553033465570345667158541759501107168465168464013834058687365.

Let's explore that thought.

Define

  1. p1 as the probability of choosing 314 from among the positive integers and
  2. p2 as the probability of choosing 1024885926414553033465570345667158541759501107168465168464013834058687365 from among the positive integers.
You have established that these probabilities differ. So, can you tell us by how much?

That is, what is the value of p1 - p2?

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I think he was just saying that you're more likely to pick a number you can handle. That very long number is not even close to long compared to more and more and more numbers... there is no average-sized number, they're just bigger and bigger and bigger.

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You're pulling my leg now, right?

I mean, how can there be several ways to do what is impossible?

LOL! :lol: Touché.

Perhaps what I should have said is that there are "various ways in which you could presume you are choosing a number at random, with equal probability to all members of the real numbers on [0,1)." Is that better? ;-D

Note that I never said it was impossible to select a random number. My contention is that it only possible to select one from a finite subset. Perhaps Chuck's statement that the probability distribution is not randomly uniform (see post 50) is a better way of viewing it.

First I chose the number of decimal places. Out of the positive integers, and at random, I chose 3.

Once you select the number of decimal places, you have narrowed the process to a finite subset. The ensuing selection may be random, but you are no longer choosing from an infinite set (because you couldn't...X-P). The only way you can ensure that all numbers occur with equal probability is if you choose an infinite number of decimal places every time you make a selection. By merely thinking of choosing a number of significant digits, you have already assured that no irrational number can ever be selected by your process. By selecting a finite number of significant digits, you have assured that you are selecting from a finite subset.

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Once you select the number of decimal places, you have narrowed the process to a finite subset.

Thank you for admitting it is possible to extract a finite subset at random from an infinite set.

You have thus contradicted your contention that such a process is impossible.

Selecting .314 at random from the reals in [0, 1] is the selection at random of a finite subset of an infinite set.

I chose the number of decimal places at random from the infinite set of positive integers.

Then I chose .314 at random from the 1000 possibilities that provided.

If you say that the choice of 3 decimal places was not a random choice, that's one thing,

but if you want others to share that belief, it requires proof.

I think your options here are three in number [and since 3 is finite, you should be able to make the choice]

  1. Prove that 3 can not have been a random selection from among the positive integers,
  2. admit that you're simply wandering around some previously unexplored territory with BrainDen as an audience, or
  3. agree that you're just pulling our collective leg. :D

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Thank you for admitting it is possible to extract a finite subset at random from an infinite set.

You have thus contradicted your contention that such a process is impossible.

Read it again. I never said you chose three decimal places at random.

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Read it again. I never said you chose three decimal places at random.

Read my post again. You have only asserted it was not random. Your task is to prove is was not at random.

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Read my post again. You have only asserted it was not random. Your task is to prove is was not at random.

Prove that you gave the number 3 an equal chance of being chosen as 10^31.

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Hi S_W,

If I understand your post, you would object to 314 as a random choice,

because you have established that the probability of choosing 314 differs

from the probability of choosing say

1024885926414553033465570345667158541759501107168465168464013834058687365.

Let's explore that thought.

Define

  1. p1 as the probability of choosing 314 from among the positive integers and
  2. p2 as the probability of choosing 1024885926414553033465570345667158541759501107168465168464013834058687365 from among the positive integers.
You have established that these probabilities differ. So, can you tell us by how much?

That is, what is the value of p1 - p2?

The answer to your question is no, I can't tell you what p1-p2 is, and it doesn't really matter. All that matters is that it's not zero. That's all. As long as it's not zero, we know that the choice of .314 isn't random because there was a higher probability of choosing 314 than 1024885926414553033465570345667158541759501107168465168464013834058687365

My reasoning is that you are far more likely to choose 314 than 73230397549047281057690760594837234859605984372556869858999685634535437242323423

2323236474859607089089684634253759706946324243769790690608080896754646353632833

because that takes way longer to type, and we all have things we'd like to do with our time other than type very large numbers, like going to the beach or having sex or going the beach while having sex.

The other issue is that I'm not really saying that the probability of choosing any number from an infinite set is zero so much as I am saying that you can't do it.

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