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Here are some questions I have and I can't ansewer

  1. Assume that we pick random a positive integer $n$. What is the probability

    (a) n=10

    (b) $n$ is even

    (c) $n$ is a prime number

  2. If we pick random a real number what is the probability this number is rational ?

Any help?

Thank's in advance!

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4  
What is a random integer? Do you assume any distribution? –  dtldarek Apr 18 '12 at 18:54
    
@dtldarek: I mean we pick random a positive integer –  passenger Apr 18 '12 at 18:55
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One cannot pick an integer at random with all integers equally likely, at least if as usual we want probabilities to be countably additive. So a distribution has to be specified. If you are interested in the densities (which are not probabilities) for $10$ we get $0$, for even we get $1/2$, for prime we get $0$. –  André Nicolas Apr 18 '12 at 18:59
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Let's say that each integer has a probability $a > 0$ of being selected. The Archimedian property states that if we take enough of these probabilities and add them up, we'll get above 1 –  The Chaz 2.0 Apr 18 '12 at 19:16
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@passenger: They are not probabilities, they are asymptotic densities. For $\{10\}$, if $n \ge 10$ we will have $|S_n|=1$ and therefore $|S_n|/n=1/n$. But $1/n\to 0$ as $n\to\infty$. For the evens, if $S_n$ is $n/2$ if $n$ is even, and $n/2-1$ if $n$ is odd. Divide by $n$. The limit is $1/2$. For the primes, I used the Prime Number Theorem, though that's overkill and one can do it with less machinery. –  André Nicolas Apr 18 '12 at 19:32

2 Answers 2

up vote 4 down vote accepted

As has been pointed out in the comments, there is no probability distribution on the positive integers that assigns equal weight to every integer.

But let $S\subseteq \mathbb{N}$ be a set of positive integers, and for every positive integer $n$, let $S_n$ be the set of all $k\in S$ such that $k\le n$. Let $|S_n|$ be the number of elements in $S_n$. Then $$\lim_{n\to\infty} \frac{|S_n|}{n},$$ if it exists, can be viewed as a measure of how "large" $S$ is. By that criterion, the answer for $S=\{10\}$ is $0$. The answer for $S$ the even numbers is $1/2$, while the answer for the primes is $0$.

However, $\lim_{n\to\infty}\frac{|S_n|}{n}$ need not exist. Moreover, even if we restrict attention to subsets of $\mathbb{N}$ for which the limit exists, this limit is not a probability distribution.

On the reals, there is no probability distribution that gives equal weight to all intervals of (say) length $1$. But let's restrict attention to a specific interval, say $[a,b]$, and use the uniform distribution on this interval. Then the probability that a randomly chosen point in $[a,b]$ is rational turns out to be $0$. Almost all real numbers are irrational, indeed almost all real numbers are transcendental.

For a proof that the rationals form a negligibly small subset of, say, $[0,1]$, let $\epsilon>0$. The rationals in $[0,1]$ form a countable set, so they can be listed as $r_1,r_2,r_3,\dots$. Put an interval of width $\frac{\epsilon}{2^1}$ about $r_1$, an interval of width $\frac{\epsilon}{2^2}$ about $r_2$, and so on. The sum of the lengths of these intervals is $\epsilon$. So the rationals are a subset of sets of arbitrarily small measure.

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$\lim_{n \to \infty} a_n$ is sometimes called the "asymptotic density", "natural density", or "arithmetic density" of $S$. –  Robert Israel Apr 18 '12 at 19:25
    
@AndréNicolas Yeah, I know that already (therefore I deleted the comment), but the part "let $S$ be a set of positive integers" can still confuse someone, "set of some positive integers" would be more clear to me, or at least please add $S \subseteq \mathbb{N}$ there. –  dtldarek Apr 18 '12 at 19:32
    
@dtldarek: Thanks, I added the $S\subseteq \mathbb{N}$ that you suggest. –  André Nicolas Apr 18 '12 at 19:34

The densities given by André Nicolas can be used as a measure of the required probabilities. For the first case the probability is zero as a limit (infinitismal probability) but when infinitely added it gives 1. This is just like the area of a strip with the width dx , the area of the strip is zero as a limit (infinitismal) but when infinitely added (the integration) it gives a value which is the area under the curve.

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