Krdytkyu

The title is my question and the reason for asking it is the following.

Define a set $mathbb N (≤ n) equiv$ {1,2,3, … , n} and define a “random selection” to be a selection in which each member of $mathbb N (≤ n)$ has an equal probability, $p$, of being selected.

As $mathbb N (≤ n)$ is the whole population from which the selection will be made then the probability of selecting some member of $mathbb N (≤ n)$ is $np = 1$.

Since $p to 0$ as $n to ∞$ then the probabilty of any member of $mathbb N $ being randomly selected is $p = 0$.

Is this correct?

Duplicate Question ?

I agree my question is similar to the other question but it is not similar enough to be called a duplicate. For example, the definition of “random selection” I use is different.

Answer from snarski

That is mostly correct. What is true is that there is no distribution on $mathbb N$ that associates to each $n in mathbb N$ an equal probability of being selected. On the other hand, that doesn't mean you can't select a random nonnegative integer "at random", because "at random" doesn't imply the distribution is uniform.

Reply to snarski

Thanks for your answer but I have some questions.

I don’t understand the statement about there being “no distribution that associates to each $n in mathbb N$ an equal probability of being selected”. Doesn’t the argument in my question show that there is such a distribution, namely $p = 0$ for each member of $mathbb N$ ?

Also I don’t understand how a member of $mathbb N$ could be randomly selected unless each member of $mathbb N$ had the same probability of selection?

The conclusion of the argument in my question seems reasonable because it implies it is impossible to make a random selection of a single member of $mathbb N$. This agrees with the practical impossibility of an algorithm that randomly selected a single member of an infinite set because an infinite set as input could not be processed by such an algorithm in a finite time.

User334732 comment

Let $f(n)=frac{1} {6}$ for all $n in mathbb N$.
Let the order be 1,2,3,4,5...
Then start with 1 and roll a dice. If you get a 6, stop and you have selected number 1 as your answer. If you don't, then go to $n$=2 and roll the dice again. Continue until you roll a 6.

Reply to user334732

With this procedure the probability of selecting $n$ is $(frac{5} {6})^{(n-1)} frac{1}{6}$ so smaller numbers have a greater probability of being selected than larger numbers. This selection does not seem to be “random”. Specifically it does not satisfy the definition of “random selection” I use in my question which is all $n in mathbb N$ must have an equal probability of being selected.

Comment from Vincent

Thanks for the clarification. What Snarski is saying is: 1) You are correct that there is no "random selection" according to your definition 2) your definition is somewhat non-standard. Usually (at least among mathematicians, the general population might lean more toward your definition) 'random' (roughly) means that probabilities are involved, not that they are necessary all equal. So yes, a random process for selecting integers that gives each equal, positive probability is impossible, but the next best thing, a random process that gives all numbers at least some positive probability, is.

user334732's comment is an example of that latter, 'next best' thing, and a very beautiful example on top of that because it is very hands on.

Reply to Vincent

It seems we agree on the impossibility of a random selection of a single member of $mathbb N$ given the definition of "random selection" in my question.

You mention that my definition is "somewhat non-standard" and while my definition may be favored by the "general population" usually to mathematicians, " 'random' (roughly) means that probabilities are involved, not that they are necessary all equal".

The definitions of a "random sample", at least the ones I have seen, say a "random sample" has only two defining properties:

P1. each member of the population has an equal chance of selection, and

P2. each selection is independent of the other selections.

My question asks about the random selection of a single member of $mathbb N$, that is, a random sample of size one from the population $mathbb N$ , so property P2 is not applicable and the general definition specializes to property P1 which is the definition in my question.

If you know of a definition of "random selection" that differs from or extends the P1 + P2 definition above then can you supply a reference ? I would be interested to read about it.

I do not share your opinion of the procedure suggested by u334732 although I appreciate u334732 taking the time to attempt to answer my question.

Problem 1 with u334732's suggestion is that the procedure does not give an equal probability of selection for each member of $mathbb N$ since the probability of selecting $n$ is $(frac{5} {6})^{(n-1)} frac{1}{6}$. Hence the procedure does not comply with the definition of "random selection" in my question. I mentioned this problem in a previous reply.

Problem 2 is that the procedure may never select a member of $mathbb N$ because the probability of not selecting a member of $mathbb N$ that is ≤ n is $(frac{5} {6})^{n}$ which is > 0 for all $n in mathbb N$ hence there is a non-zero probability that no member of $mathbb N$
will be selected by any finite number of applications of the procedure.

I think these problems reflect the impossibility of randomly selecting a single member of $mathbb N$ and there would be some problem with any suggested procedure.

That is mostly correct. What is true is that there is no distribution on $mathbb{N}$ that associates to each $n in mathbb{N}$ an equal probability of being selected. On the other hand, that doesn't mean you can't select a random nonnegative integer "at random", because "at random" doesn't imply the distribution is uniform.