Which is the probability to a random line to be parallel to a specific other line? In my perception, using the common sense, is less common, or less probable, to a random line be parallel that not to be, because to be parallel a line needs obey a restrictive rule. But anyone can, using simple probability, obtain that result:
specific line (r) = any line in R2 (e.g. y=0)
amount of random lines (u) = infinite (inf)
amount of parallel lines to r (s) = infinite (inf)
probability of s parallel to r = s / u = inf/inf = undefined

But there is another solution? Maybe using geometric probability, integrals or even empirical results?
[EDIT]
OK. I've understood the zero result and the "not impossible" thing. Thanks for answers and references.
But... 


*

*Is my initial perception wrong? 

*Is not easier to find a not parallel line instead a parallel? 

*Could someone prove or negate it?

*If my perception is not wrong someone can calculate how easier is?

*or maybe I really didnt understand the answers?


[NEW EDIT]
So, is it the final answer?
- Is my initial perception wrong? 
- A: No, its correct.


*

*Is not easier to find a not parallel line instead a parallel? 

*A: Yes, it is easier.

*Could someone prove or negate it?

*A: Yes: '''The probability of finding a parallel line is zero, so the probability of finding a non-parallel line is equal to one. Since 1>0, you have the answer to your question. – Rod Carvalho'''

*If my perception is not wrong someone can calculate how easier is?

*A: No, nobody can because it is undefined. (?)


[FINAL EDITION]
Now I realize that the question resumes to: which is the probability of a random real number be equal to a specific other. Thanks for help.
 A: Suppose that we have a line $\mathcal{L}_1$ defined by 
$$\mathcal{L}_1 := \{ (x,y) \in \mathbb{R}^2 \mid y = a x + b \}$$
and another line, $\mathcal{L}_2$, defined by
$$\mathcal{L}_2 := \{ (x,y) \in \mathbb{R}^2 \mid y = \tan (\theta) x + c \}$$
where $\theta$ is an observation of a random variable $\Theta$ uniformly distributed over $[0, \pi]$. Lines $\mathcal{L}_1$ and $\mathcal{L}_2$ will be parallel if $\theta = \tan^{-1} (a)$. However, since $\Theta$ has a continuous distribution, we have that 
$$\mathbb{P} \left( \Theta = \tan^{-1} (a) \right) = 0$$
In other words, the probability that a line whose slope is randomly chosen  is parallel to a given line is exactly equal to zero.
A: A line in $\Bbb R^2$ is parallel to another line if and only if their slopes are equal, regardless of their intercept.
The probability that the slope $m$ of a random line is equal to the slope $m_f$ of some fixed line is therefore $P(m = m_f) = 0$, since $m_f$ is continuously distributed.
A: The answer depends on the random distribution accordingto which the lines are produced. But if the probability of a random line being parallel to a given line $L$ is assumed not to depend on $L$ (a resonable symmetry assumption), then the assumption of a positive probability $p>0$ leads to a contradiction:
If $p>0$ then there exists a natural number $n$ such that $n>\frac1p$.
It is easy to construct $n$ differnt lines $L_1,\ldots, L_n$ through a single point $A$. The probability of a random line $L$ to be parallel to $L_i$ is assumed to be $p$ for all $i$. Since the events $L||L_i$ and $L||L_j$ are mutually exclusive, we obtain that the probability that $L$ is parallel to any of the $L_i$ is $p+p+\ldots+p$ with $n$ summands. This produces a total probability of $np>1$ - contradiction.
We conclude that the assumption $p>0$ is wrong, hence $p=0$.
