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Suppose I can generate numbers uniformly at random from an infinite set, such as:

$$r \in \mathbb{R} : 0 < r < 1$$

Each number has an infinitely small probability of being generated. Does that mean any two randomly generated $r$ are guaranteed to be unique?

I think this is a weaker version of the same question: Suppose an algorithm generates two bits uniformly at random. If the two bits are different, it starts over. Otherwise, it halts.

Is the algorithm guaranteed to halt?

(Assuming the universe can provide infinite entropy/has an infinite lifespan.)

mgiuffrida
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2 Answers2

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If each generation is an independent event (and in this case I think we can assume it is), then no.

The outcome of one generation does not affect the outcome of any other generation (independence), so the outcome of generating a particular number $r$ has the same non-zero probability each time you generate a random number.

Adam Francey
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  • Good point, but that still seems like a contradiction when the probability of any outcome $r$ is infinitely small. If each bit of two numbers is generated randomly and there are an infinite number of bits, won't a bit eventually differ? – mgiuffrida Jun 15 '16 at 17:46
  • If have a collection of objects and I pick one at random and put it back. I pick another one at random from the same collection, what's stopping me from picking the same object again? Nothing, even if there are an infinite amount of objects in the collection.

    The question in your comment is kind of like asking, "If I flip a coin an infinite amount of times, won't it eventually land on tails?". Not necessarily!

     – Adam Francey Jun 15 '16 at 18:59
  • Not necessarily? If I flip a coin an infinite number of times: $P(\text{numTails} > 0) = \lim_{n \to \infty}(1 - P(\text{numHeads = n}) = \lim_{n \to \infty}(1 - .5^n)$ which is $1$. – mgiuffrida Jun 16 '16 at 05:40
  • Additionally, is the probability of generating a particular $r$ non-zero? $P(r) = \lim_{n \to \infty}(1/n)$ which seems like a paradox. – mgiuffrida Jun 16 '16 at 05:47
  • The probability is one, but that only means it happens almost surely. – Adam Francey Jun 21 '16 at 02:23
  • @mgiuffrida The probability of generating a particular $r$ is non-zero here, but only because you stated that each has an "infinitely small probability of being generated". In actuality, your question goes deeper because there is no uniform distribution over the real numbers. If there were a uniform distribution, and it assigned a probability of 0 to every event, I assume there would would a lot of problems. – Adam Francey Jun 21 '16 at 02:33
  • Interesting, found some more discussion here about uniform distributions over the real numbers. So my premise is basically a paradox, at least as far as I can fathom. – mgiuffrida Jun 21 '16 at 07:37
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Assuming a uniform PDF, choose $x_0$ and then $x_1$ from $(0,1).$ For every $n\in N,$ the probability that $|x_1-x_0|<1/n,$ which is at most $2/n,$ is greater than or equal to the probability that $x_1=x_0.$ So the probability that $x_1=x_0$ is $0.$

What is the precise mathematical meaning of "guaranteed to be unique" ?

DanielWainfleet
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