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Given a random integer, is the probability of correctly guessing what it is exactly zero? What if it would be a real number, rather than an integer? Does the fact that the set of all integers is countable and the set of real numbers is uncountable change the probability value?

Constantine
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    When you say, "Given a random integer," how exactly are you picking randomly from the integers? There isn't a uniform probability distribution on them. – Potato Jun 22 '13 at 20:39

2 Answers2

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Actually, there is something pleasantly deep about this question.

Firstly, it simply doesn't make sense to talk about taking a 'random integer', if by random you mean that every integer has an equal chance to be taken. In other terms, there is no countably infinite uniform probability distribution. Why? The typical model for probability relies on

  1. non-negative probability of any set of events
  2. the total probability = 1
  3. countable additivity of disjoint events

But you can't have countably infinitely many probabilities that both sum to 1 and are equally likely.

On the other hand, it's not so hard to put a uniform-measure on $[0,1]$ (which is in bijection with the reals) that describes a process of randomly choosing real numbers, and each real number has probability zero of being chosen.

davidlowryduda
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  • @user: I do not understand your objection. I do not attempt to explain that at all, because I merely refer to the standard axioms of Probability. – davidlowryduda Jun 23 '13 at 03:05
  • @user: I still do not understand what you are saying to me. Did you read the linked page under that comment? I did not mention finite additivity at all. Fortunately, this problem is countably infinite, so I did not need to mention finite additivity at all. I mean this and exactly this: you cannot talk about a random integer if by random you mean every integer has an equal chance to be taken. The reason why is because it violates the axioms of probability. – davidlowryduda Jun 23 '13 at 07:29
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There really isn't such a thing as a random integer.

Or more precisely: there are a lot of ways to define "random integer", and none of them are particularly good. You can in fact show that there is no way to define "random integer" so that the chance that the randomly selected integer is divisible by $m$ is $1/m$ for any $m > 1$.

For integers, to how you select a random integer, you essentially have to give for each integer $n$ probability of selecting $n$, say $p_n$. Of course, you need $\sum_n p_n = 1$. So, you may have the probability of selecting the "guessed number" equal to any real number in $[0,1]$, with no value distinguished.

For reals, the situation is more subtle. You again have no distinguished probability, and to describe any particular one, you have to give a more subtle object called probability measure (as opposed to just a bunch of random numbers $p_n$). You can have a continuous proability measure, where the chance of selecting $x \in A$ is $\int_A f(x) dx$ for some "nice" function $f$ called density. You can have a discrete distribution, where you say that a value $x_n$ has probability $p_n$ of being selected. And you can have much more subtle behaviour. Again, proablitily of selecting $0$ may or may not be $0$.

Jakub Konieczny
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