Can there be a lottery of the natural numbers, so that every natural number is chosen equally likely?

The standard answer would be "No" because: If we define a measure $\mathbf{P}$ on $\mathbb{N}$ so that $\mathbf{P}(n) = r \in (0,1] \; \forall \, \mathbb{N}$, then $\mathbf{P}(\mathbb{N}) = \infty$. If we define a measure so that $\mathbf{P}(n) = 0$, then $\mathbf{P}(\mathbb{N}) = 0$.

But why can we conclude from that, that a lottery of the natural numbers (with every natural number equally likely) is impossible?

Note: the question is not if there can be a uniform probability distribution (satisfying all axioms of probability, including countable additivity) over the natural numbers but if there can be a lottery of the natural numbers so that every number is chosen equally likely!


marked as duplicate by Did, Charles, jameselmore, Rory Daulton, Alex Provost Dec 30 '15 at 20:25

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  • $\begingroup$ You could make $p(n)=1/2^n$. $\endgroup$ – Gregory Grant Nov 25 '15 at 14:42
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    $\begingroup$ I don't understand what you mean by "If we define a measure so that $P(n)=0$ then $P(\Bbb N)=1$.". $\endgroup$ – Gregory Grant Nov 25 '15 at 14:44
  • $\begingroup$ @GregoryGrant I suspect that was a typo. I repaired it in an edit. Let the OP check it. $\endgroup$ – drhab Nov 25 '15 at 14:46
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    $\begingroup$ No, there is no way around it. There is no uniform (probability) measure on a countably infinite set. When one needs something like this, a typical approach is to obtain a result for a uniform probability on a finite set $\{1,\ldots, N\}$ and then pass to a limit as $N \to \infty$, depending on what actually is to be shown. For example, one might ask: what is the probability that "two natural numbers chosen at random" are coprime? A sensible interpretation of this question can be made using the limit approach, although strictly speaking there is no way to uniformly select a natural number. $\endgroup$ – hardmath Nov 25 '15 at 14:47
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    $\begingroup$ If you want each integer to have an identical probability such that the probabilities sum to $1$, then at the very least you'll need to work over a non-archimedian field. Maybe a solution to this exists in non-standard analysis. $\endgroup$ – Omnomnomnom Nov 25 '15 at 14:50

The probability that some number is drawn cannot be positive because the sum of the probabilities would be infinite.

Probability $0$ for an event not impossible is possible, if the number of events is uncountable. But for countably many events, $P(X)=0$ is equivalent to $X$ is the impossible event.

  • $\begingroup$ Where is the flaw in the following: 1.) I choose randomly on $[0, 1)$ with uniform distribution until I get a result in $A = \mathbb{Q} \cap [0, 1)$ (rationals are spreaded equally in the reals) 2.) I use a bijection $\phi\colon A \rightarrow \mathbb{N} $, voilà, I have chosen randomly a natural number (with all of them equally likely)! $\endgroup$ – R. Neville Nov 30 '15 at 5:59
  • $\begingroup$ The probability that you ever choose a rational number in the interval $[0,1)$ is $0$ due the the uncountability of the reals $\endgroup$ – ASKASK Nov 30 '15 at 6:07
  • $\begingroup$ So the whole "choose randomly until I get a rational result" will have you never reaching an answer $\endgroup$ – ASKASK Nov 30 '15 at 6:08
  • $\begingroup$ @ASKASK: Yes, I know $\mathbb{P}(A) = 0$, but it is still possible! After all, if I chose a real number $r$ in $[0,1)$ it was $\mathbb{P}(r) = 0$, but still it happened! $\endgroup$ – R. Neville Nov 30 '15 at 6:14
  • $\begingroup$ To be completely honest, I have very little knowledge on this subject so feel free to ignore my answer as it may be wrong, but from how I see it, you can't simply rely on the notion of "inevitably landing on a rational" because there is nothing that guarentees that it will actually ever happen (in fact, odds are it just never happens) $\endgroup$ – ASKASK Nov 30 '15 at 6:19

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