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unreality

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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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I have another :D

5) You use your infinite-random-digit generator from #4 to create an infinite stream of random digits. What are the chances of generating the decimal part of pi (ie, disregarding the 3 in front, and just focusing on the part following the decimal)

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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?

As long as a "random" number is really random, the chance of getting the conditions must be greater than 0%, only fractionally greater but greater none the less.

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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

In [1] and [2] I read we pick positive or non-negative integers.

In [3] I pick a random real number - rational or irrational - from the interval [0, 1].

In [4] and [5] it's clear.

  1. 0.
  2. 0.
  3. 0, 0, 0.
  4. 0.
  5. 0.

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Actually, this is a very interesting problem, since the answer is

0 but not really 0, actually, the term is "infinitesimal"

I like limits, so I'd probably put the answers in terms of limit as x->infinity of any of those situations...

I know that it is actually a hot topic in the field of mathematics, but I haven't had that much experience with it...I had a friend who worked on measure theory and he would probably know...I may ask him and post again :P

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I'm not as great with numbers as many of the people here, but I thought it would be
1/infinite

0.

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1.) you misused the word "integer" but whatever, its not relevant (integers include 0 negative numbers like -12). The answer to this questions is, obviously, 1 divided by infinity, which in this case we can safely call zero, but if you're really adventurous you might call "mathematically undefined" (sort of like 5/0). You might reply that there has to be some probability that you will choose 11,734, but since you could never actually perform this experiment (even theoretically), there really isn't any probability that you will choose 11,734.

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The answer to all of these questions is 0 or 1/infinite. The actual answer would be 0.000000000000000000000000000000000000000000000000000000000000000000000000000000

00000000000000000000000000000000.... etc... without any real end, that's the reason why it's just used as 0, it's the nearest approximation.

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Heya!

(For part 1): I thought that anything on an infinite time scale was certain to happen- i.e the probability is 1 or 100%. Clearly this isn't correct though, so please would somebody explain to me why.

Thanks :)

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only one person got it completely right ;D

1/infinity is not NECESSARILY zero, but is usually defined as infintesimal

congrats Yoruichi!

there is the slightest probability that you could pick those numbers, just HIGHLY UNPROBABLE. 1/infinity unprobable. The interesting thing with the first two questions that I wanted to point out was that they are both 1/inf, yet the second one has MORE numbers involved, so that should be a harder probability? it's confusing to think about, lol

and is the probability to get the decimals of the golden ratio 'harder' to get than the decimals of pi? hmmm...

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only one person got it completely right ;D

1/infinity is not NECESSARILY zero, but is usually defined as infintesimal

congrats Yoruichi!

there is the slightest probability that you could pick those numbers, just HIGHLY UNPROBABLE. 1/infinity unprobable. The interesting thing with the first two questions that I wanted to point out was that they are both 1/inf, yet the second one has MORE numbers involved, so that should be a harder probability? it's confusing to think about, lol

and is the probability to get the decimals of the golden ratio 'harder' to get than the decimals of pi? hmmm...

Dude, infintesimal is 0. When dealing with the infinite, things don't work exactly like you think they normally would. I've read books on the matter, trust me, 1/infinity = 0. Think about it this way. Say you have 1/1. Then start to increase the denominaor. As the denominator increases, the number gets smaller and smaller, becoming more ifintesimal, like you said. But as it goes to infinity, the actual value goes to zero. The answer is zero, especially if you think about it with limits.

Oh, and more numbers doesn't affect the probability. It's the infinite. Something else interesting. If you think about the set of natural numbers(1,2,3,4...) and the set of primes(1,4,9,16...), you can set up a one-to-one correspondance between them(1-1,2-4,3-9,4-16...). This makes it seem like there are as many primes as natural numbers, but the primes are a subset of the natural numbers! The two sets have the same cardinality. Infinity can get weird at times.

Edited by Frost

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Think about it this way: a probability is, by definition, a real number in the interval [0,1]. If the probability of these events is anything other than 0, what is it? It must be a number.

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Dude, infintesimal is 0. When dealing with the infinite, things don't work exactly like you think they normally would. I've read books on the matter, trust me, 1/infinity = 0. Think about it this way. Say you have 1/1. Then start to increase the denominaor. As the denominator increases, the number gets smaller and smaller, becoming more ifintesimal, like you said. But as it goes to infinity, the actual value goes to zero. The answer is zero, especially if you think about it with limits.

Oh, and more numbers doesn't affect the probability. It's the infinite. Something else interesting. If you think about the set of natural numbers(1,2,3,4...) and the set of primes(1,4,9,16...), you can set up a one-to-one correspondance between them(1-1,2-4,3-9,4-16...). This makes it seem like there are as many primes as natural numbers, but the primes are a subset of the natural numbers! The two sets have the same cardinality. Infinity can get weird at times.

Frost: I think you need to look at this.

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Frost: I think you need to look at this.

I just did. You have a point, but he mentions limits in there. As it goes to infinity, the number approaches zero. You may call it infintesimal, but I would consider that the same thing as saying it equals zero. limx->infinity(1/x) = 0

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I just did. You have a point, but he mentions limits in there. As it goes to infinity, the number approaches zero. You may call it infintesimal, but I would consider that the same thing as saying it equals zero. limx->infinity(1/x) = 0

"Approaches zero" and "equal to zero" are not the same thing. As an example, what is the smallest number in the open interval (0,1)? It can't be zero, which is explicitly excluded.

Chuck is correct in saying it is "almost surely" zero. In this case, we are speaking probability, so it means that, while the event is possible, it is not only unlikely, but expected never to occur. Thus, if we are pressed to give a numerical answer, we can confidently say zero.

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"Approaches zero" and "equal to zero" are not the same thing. As an example, what is the smallest number in the open interval (0,1)? It can't be zero, which is explicitly excluded.

Chuck is correct in saying it is "almost surely" zero. In this case, we are speaking probability, so it means that, while the event is possible, it is not only unlikely, but expected never to occur. Thus, if we are pressed to give a numerical answer, we can confidently say zero.

I see your point. :D

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How can you? It is infinitesimally small! :D

lol, well, it's somewhere, i just...can't...reach it! :lol:

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hehe ;D as others have said "approaches 0 as a limit" does not mean "equals zero". It is a class of negative-powered cardinals called 'infintesimals', they're part of the aleph ladder (sort of)... ie, the reciprocals of the alephs. Infintesimal. Infinitely small, but still something, just something so small it's infinitely small ;D impossible to grasp :P

therefore, we can say the probability of picking 11,734 is infinitely small, but NOT ZERO. Think about it. If every number had a probability of 0 of being picked, NONE OF THE NUMBERS WOULD BE PICKED. Since one number is picked, the answer is not 0. It's "infintesimal" :P

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i have come to terms with my inability to cope with the infinite mass/ space/ time/ volume or number......

:P

gonna leave these puzzles alone before they hurt me...

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hehe ;D as others have said "approaches 0 as a limit" does not mean "equals zero". It is a class of negative-powered cardinals called 'infintesimals', they're part of the aleph ladder (sort of)... ie, the reciprocals of the alephs. Infintesimal. Infinitely small, but still something, just something so small it's infinitely small ;D impossible to grasp :P

therefore, we can say the probability of picking 11,734 is infinitely small, but NOT ZERO. Think about it. If every number had a probability of 0 of being picked, NONE OF THE NUMBERS WOULD BE PICKED. Since one number is picked, the answer is not 0. It's "infintesimal" :P

From wikipedia, the font of all knowledge: "The naive definition of an infinitesimal is this: a number whose absolute value is less than any non-zero positive number. From this definition, it can be shown than there are no non-zero real infinitesimals, using the property of the least upper bound. Considering just the positive numbers, the only way for a number to be less than all numbers would be to be the least positive number. If h is such a number, then what is h/2? Or if h is indivisible, is it still a number?"

http://en.wikipedia.org/wiki/Infinitesimal

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