Randomly selecting a natural number In the answer to these questions:


*

*Probability of picking a random natural number, 

*Given two randomly chosen natural numbers, what is the probability that the second is greater than the first? 
it is stated that one cannot pick a natural number randomly.
However, in this question:


*

*What is the probability of randomly selecting $ n $ natural numbers, all pairwise coprime?
it is assumed that we can pick $n$ natural numbers randomly.
A description is given in the last question as to how these numbers are randomly selected, to which there seems to be no objection (although the accepted answer is given by one of the people explaining that one cannot pick a random number in the first question).
I know one can't pick a natural number randomly, so how come there doesn't seem to be a problem with randomly picking a number in the last question?
NB: I am happy with some sort of measure-theoretic answer, hence the probability-theory tag, but I think for accessibility to other people a more basic description would be preferable.
 A: It really depends on what you mean by the "probability of randomly selecting n natural numbers with property $P$". While you cannot pick random natural number, you can speak of uniform distribution.
For the last problem, the probability is calculated, and is to be understood as the limit when $N \to \infty$ from the  "probability of randomly selecting n natural numbers from $1$ to $N$, all pairwise coprime".
Note that in this sense, the second problem also has an answer. And some of this type of probabilities can be connected via dynamical systems to an ergodic measure and an ergodic theorem. 

Added The example provided by James Fennell is good to understand the last paragraph above.
Consider ${\mathbb Z}_2 = {\mathbb Z}/2{\mathbb Z}$, and the action of ${\mathbb Z}$ on ${\mathbb Z}_2$ defined by
$$m+ ( n \mod 2)=(n+m) \mod 2$$
Then, there exists an unique ergodic measure on ${\mathbb Z}_2$, namely $P(0 \mod 2)= P(1 \mod 2)= \frac{1}{2}$.
This is really what we intuitively understand by "half of the integers are even".
Now, the ergodic theory yields (and is something which can be easily proven directly in this case)
$$\lim_{N} \frac{\text{amount of even natural numbers} \leq N}{N} = P( 0 \mod 2) =\frac{1}{2} \,.$$
A: Perhaps a justification is this. In the first question it is (correctly) claimed that it is impossible to have a uniform distribution on the natural numbers. Thus we can't develop a sensible way of choosing particular numbers at random, when each supposedly has the same probability. The last question though is dealing with the probability of picking a certain class of numbers. In that case the approach is to pick $N$ large, impose a uniform distribution on $[1,N]$ (which we can always do), work out the probability as a function of $N$, and then take the limit $N \rightarrow \infty.$
Example: what's the probability of randomly picking an even number?
Fix $N$. If $N$ is even then the probability of picking an even number in the uniform distribution on $[1,N]$ is exactly $1/2$. If $N$ is odd then the probability is
$$
\frac{\text{amount of even numbers}}{\text{total amount}} = \frac{ (N-1)/2 }{N} = \frac{1}{2} - \frac{1}{2N}
$$
And so, as we take the limit $N \rightarrow \infty$, we say that the probability of choosing an even number is 1/2.

This approach is really a "natural density" approach; see http://en.wikipedia.org/wiki/Natural_density. 
The natural density features in, for instance, the Green-Tao theorem. Green and Tao showed that the primes have positive natural density, and thus that they must contain arbitrarily long arithmetic sequences. <- incorrect!!
A: By partitioning the real numbers using non-measurable sets, it is possible to relate a real selected uniformly at random from an interval like [0, 1) to a natural number using a method by which all natural numbers have an "equal" chance of being mapped to (here, equal simply means that no natural is given any sort of preference as to being selected over any other natural), however, the probability of picking any given natural number is undefined as a result of having to partition the reals into non-measurable sets in order to create the mapping.  As a result, we are not able to create a cumulative distribution function that is meaningful.  To see an example of this, see the following question:
Is this fraction undefined? Infinite Probability Question.
A: There is no uniform distribution on the natural numbers. 
