Probability that one number divides another of two randomly chosen numbers I've met the following problem:
There's a set of numbers $\{1, 2, 3, ..., 2n\}$. After that we randomly choose two numbers $a$ and $b$. What's the probability that $a$ divides $b$ or $b$ divides $a$.

An expression for the number of pairs $(a,b)\in\{1,\dots,m\}^2$ that satisfy $[a\mid b\text{ or }b\mid a]$ is:$$2\sum_{r=1}^m\lfloor\frac{m}{r}\rfloor-m$$
For every $r\in\{1,\dots,m\}$ there are $\lfloor\frac{m}{r}\rfloor$ multiples of $r$ in the set $\{1,\dots,m\}$. 
Taking $\lfloor\frac{m}{r}\rfloor$ twice we get the number of pairs plus $m$ (because pairs on the diagonal are counted twice).
So an expression for the probability that a pair satisfies the condition is:$$\left[2\sum_{r=1}^m\lfloor\frac{m}{r}\rfloor-m\right]/m^2$$
In your case we must substitute $m=2n$.
