probability picking random a positive integer Here are some questions I have and I can't ansewer 


*

*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

*If we pick random  a real number what is the probability this number is rational ?
Any help?
Thank's in advance!
 A: 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.
A: 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.
