7 ms·

Another "paradox": even though it's possible to randomly pick a rational number from the reals, the probability of this happening is 0.

Sinergy2 9y ago
Please describe how it is possible to pick such a number. For example, I can readily imagine how to pick a random 32b float, but that it is an entirely problem with a nonzero probability.
- Sinergy2 9y ago
  *entirely different
- georgewsinger 9y ago
  Usually in math we assume the axiom of choice :) https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice I'm assuming this could somehow lead to such a "random" pick in the technical sense. In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer. Perhaps a mathematician could show how to pick one on some abstract machine with infinite resources, and not constrained by finite bit representations of numbers.
  - red75prime 9y ago
    Run a random number generator (0-9) for each decimal position after the dot in parallel. This should do a trick.
  - dragonwriter 9y ago
    > In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer An actual (finite in time and space) computer can't even represent arbitrary real numbers, much less randomly choose them.
    - SomeStupidPoint 9y ago
      A Turing machine can't pick random numbers of any kind. Once you accept that you have an entropy source in the physical world, you can easily be injecting random real numbers (from some range) and in fact, usually are, which are then being binned into integers by ADCs.
    - dorgo 9y ago
      What about PI? We can represent it in terms of "we know what we are talking about" and we can distinguish it from other numbers.
      - leni536 9y ago
        You can only have countable number of first or second order logic statements each defining a specific real number.
    - ouid 9y ago
      Those things sound equivalent, but I don't want to be the one to try to prove it.
- stiff 9y ago
  In probability theory, when dealing with continuous sample spaces / random variables, events with probability 0 still have a chance of occurring, and events with probability 1 stil l have a chance of NOT occurring, see: https://en.wikipedia.org/wiki/Almost_surely https://en.wikipedia.org/wiki/Almost_surely This strange property comes from strange properties of the real numbers (and uncountably infinite sets) that give rise to things like: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox Measure theory deals with resolving this: https://en.wikipedia.org/wiki/Measure_(mathematics) https://en.wikipedia.org/wiki/Measure_(mathematics)
  - ouid 9y ago
    What's an anagram for Banach Tarski? Banach Tarski Banach Tarski. Seriously though, you can do math without invoking the axiom of choice. The formulation of probability doesn't strictly depend on it.
btilly 9y ago
There is no uniform probability distribution on the reals. Perhaps you meant the interval from 0 to 1?
- Mattasher 9y ago
  There are many ways to pick randomly from the rationals with a non-uniform distribution.
ramanan 9y ago
there are a number of similar paradoxes that arise when considering infinities of different sizes! Infinity Paradoxes - Numberphile - https://www.youtube.com/watch?v=dDl7g_2x74Q https://www.youtube.com/watch?v=dDl7g_2x74Q
philipov 9y ago
I think this only sounds like a paradox if it is phrased poorly. The accurate way to state it is "The probability of randomly picking a specific number is 0" and that sounds reasonable. The probability of successfully picking any number is 1.
- cdavid 9y ago
  That's a different statement: OP is alluding to the fact the measure of Q is 0 when using the "standard" sigma algebra on the real line, while you are saying that the measure of a number of 0. [edit] strictly speaking, you would restrict yourself to a bounded interval, e.g. if you pick a random number from a uniform distribution on [0, 1], the probability that this number is rational is 0.
  - philipov 9y ago
    oh, yeah, but that's because although Q is dense, it is not a dense subset of R and locally that's equivalent to saying a single point is not dense in R
    - cdavid 9y ago
      no, it is not. The OP statement is about the probability of the event "{X in Q}" (equal to 0), with X a random variable uniformly distributed on a bounded interval. That even contains many points (infinitely many actually), but has a probability 0. You are talking about the probability of a single point event, which is also always 0 on that same sigma algebra. The OP point is not completely trivial because the event contains an infinite (but countably) number of elements. It is fairly easy to understand though since by its very definition, the P[{X in Q}] = sum P[{x}] taken over every rational number (since Q is countable), and each P[{x}] is 0. A deeper statement is that there exists uncountable sets of probability 0.
- Houshalter 9y ago
  The paradox is that, after picking a random number, you have just done a thing which has probability zero. Doing a thing that has zero probability shouldn't be possible. Ever.
cdavid 9y ago
I find that result fairly intuitive, when you understand how measure theory came up to be. A much more surprising result is that most irrational are normal numbers, but we know almost no normal number (morally speaking, a normal number is an irrational number where each digit is equiprobable in any base).
- waqf 9y ago
  Define "almost no". I mean, they have full measure and I can give countably many explicit examples, what more could you expect?
  - ouid 9y ago
    I think all mathematicians, given enough time, will eventually say countable to mean infinite.
BeetleB 9y ago
This is virtually an axiom for continuous distributions. One of the axioms of probability is that if you have an event (i.e. a set), then the probability of a countable union of disjoint sets is the sum of the probability of each set (event) occurring. Assume a uniform distribution between 0 and 1. Now consider point sets of the rationals (i.e. the number 0.5 is represented by a set with just 0.5 in it). Since the distribution is uniform, each set has the same probability (i.e. the likelihood of picking a random rational). Now consider this question: What is the probability of picking any rational between 0 and 1? Well, that's just the sum of the probabilities over all rationals (because it is a countable sum of disjoint sets). If the probability of picking any particular rational was non-zero, this sum would be infinite, which violates the laws of probability. Thus, by convention, it's just simpler to define it to be 0. There's no magic here. These properties were picked merely to make analysis with measure theory clean. Don't try to ascribe any real world meaning to picking a point.

Sinergy2 9y ago

Please describe how it is possible to pick such a number. For example, I can readily imagine how to pick a random 32b float, but that it is an entirely problem with a nonzero probability.

Sinergy2 9y ago
*entirely different
georgewsinger 9y ago
Usually in math we assume the axiom of choice :) https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_choice I'm assuming this could somehow lead to such a "random" pick in the technical sense. In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer. Perhaps a mathematician could show how to pick one on some abstract machine with infinite resources, and not constrained by finite bit representations of numbers.
- red75prime 9y ago
  Run a random number generator (0-9) for each decimal position after the dot in parallel. This should do a trick.
- dragonwriter 9y ago
  > In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer An actual (finite in time and space) computer can't even represent arbitrary real numbers, much less randomly choose them.
  - SomeStupidPoint 9y ago
    A Turing machine can't pick random numbers of any kind. Once you accept that you have an entropy source in the physical world, you can easily be injecting random real numbers (from some range) and in fact, usually are, which are then being binned into integers by ADCs.
  - dorgo 9y ago
    What about PI? We can represent it in terms of "we know what we are talking about" and we can distinguish it from other numbers.
    - leni536 9y ago
      You can only have countable number of first or second order logic statements each defining a specific real number.
  - ouid 9y ago
    Those things sound equivalent, but I don't want to be the one to try to prove it.
stiff 9y ago
In probability theory, when dealing with continuous sample spaces / random variables, events with probability 0 still have a chance of occurring, and events with probability 1 stil l have a chance of NOT occurring, see: https://en.wikipedia.org/wiki/Almost_surely https://en.wikipedia.org/wiki/Almost_surely This strange property comes from strange properties of the real numbers (and uncountably infinite sets) that give rise to things like: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox Measure theory deals with resolving this: https://en.wikipedia.org/wiki/Measure_(mathematics) https://en.wikipedia.org/wiki/Measure_(mathematics)
- ouid 9y ago
  What's an anagram for Banach Tarski? Banach Tarski Banach Tarski. Seriously though, you can do math without invoking the axiom of choice. The formulation of probability doesn't strictly depend on it.

btilly 9y ago

There is no uniform probability distribution on the reals. Perhaps you meant the interval from 0 to 1?

Mattasher 9y ago
There are many ways to pick randomly from the rationals with a non-uniform distribution.

ramanan 9y ago

there are a number of similar paradoxes that arise when considering infinities of different sizes! Infinity Paradoxes - Numberphile - https://www.youtube.com/watch?v=dDl7g_2x74Q https://www.youtube.com/watch?v=dDl7g_2x74Q

philipov 9y ago

I think this only sounds like a paradox if it is phrased poorly. The accurate way to state it is "The probability of randomly picking a specific number is 0" and that sounds reasonable. The probability of successfully picking any number is 1.

cdavid 9y ago
That's a different statement: OP is alluding to the fact the measure of Q is 0 when using the "standard" sigma algebra on the real line, while you are saying that the measure of a number of 0. [edit] strictly speaking, you would restrict yourself to a bounded interval, e.g. if you pick a random number from a uniform distribution on [0, 1], the probability that this number is rational is 0.
- philipov 9y ago
  oh, yeah, but that's because although Q is dense, it is not a dense subset of R and locally that's equivalent to saying a single point is not dense in R
  - cdavid 9y ago
    no, it is not. The OP statement is about the probability of the event "{X in Q}" (equal to 0), with X a random variable uniformly distributed on a bounded interval. That even contains many points (infinitely many actually), but has a probability 0. You are talking about the probability of a single point event, which is also always 0 on that same sigma algebra. The OP point is not completely trivial because the event contains an infinite (but countably) number of elements. It is fairly easy to understand though since by its very definition, the P[{X in Q}] = sum P[{x}] taken over every rational number (since Q is countable), and each P[{x}] is 0. A deeper statement is that there exists uncountable sets of probability 0.
Houshalter 9y ago
The paradox is that, after picking a random number, you have just done a thing which has probability zero. Doing a thing that has zero probability shouldn't be possible. Ever.

cdavid 9y ago

I find that result fairly intuitive, when you understand how measure theory came up to be. A much more surprising result is that most irrational are normal numbers, but we know almost no normal number (morally speaking, a normal number is an irrational number where each digit is equiprobable in any base).

waqf 9y ago
Define "almost no". I mean, they have full measure and I can give countably many explicit examples, what more could you expect?
- ouid 9y ago
  I think all mathematicians, given enough time, will eventually say countable to mean infinite.

BeetleB 9y ago

This is virtually an axiom for continuous distributions. One of the axioms of probability is that if you have an event (i.e. a set), then the probability of a countable union of disjoint sets is the sum of the probability of each set (event) occurring. Assume a uniform distribution between 0 and 1. Now consider point sets of the rationals (i.e. the number 0.5 is represented by a set with just 0.5 in it). Since the distribution is uniform, each set has the same probability (i.e. the likelihood of picking a random rational). Now consider this question: What is the probability of picking any rational between 0 and 1? Well, that's just the sum of the probabilities over all rationals (because it is a countable sum of disjoint sets). If the probability of picking any particular rational was non-zero, this sum would be infinite, which violates the laws of probability. Thus, by convention, it's just simpler to define it to be 0. There's no magic here. These properties were picked merely to make analysis with measure theory clean. Don't try to ascribe any real world meaning to picking a point.