A line with a random length, attaching to the origin. What is the chance of a given point along 1D axis belonging to that line? The question is based on an engineering problem and, due to my lack of knowledge in statistics, I found it difficult to express the question clearly in maths language. 
The problem is formulated like this:
1) Imagine there is an origin, and a 1D coordinate system (say, L);
2) Along this 1D axis (L), there is a 'rigid' rod attaching to the origin, and its length is a random variable (always positive) and follows a PDF p1.
Question 1: for each given point on the axis L, what is the chance of that point belonging to the thread. 
My solution to Q1: for a given coordinate x, for it to 'fall' into the thread, the thread length (a) has to be equal or greater than x. Therefore, the chance of a point x belonging to that rod is equal to the integration of that PDF, from x to infinity.
Can anyone please confirm if my solution above is correct?
If so, the question 2 is as follows: if the near end of the rod no longer attaches to the origin, but the location of it is also a random variable, which follows, say, another, PDF p2. Then, for each given point on the axis L, what is the chance of that point belonging to the rod?
I am so confused by this question 2, as I have no idea what I should do with the potential interplay between these two random events (i.e. the length of rod and the location of near end of the rod). I wonder if anyone here can help me with this, or at least let me know what kind of 'theory' I should look into for solving this problem?
 A: The terminology of probability--outcomes, events, and random variables--was invented to help with such problems.  But let's begin with a picture of the situation:

This figure plots possible values of the endpoints $(X,Y).$  They are random variables.  The axes are copies of the line; its origin is at the bottom left where the axes meet.
Pick a point $x$ on the line, as shown simultaneously on both axes.  


*

*Question 1 concerns the event 
$$\mathcal{E}_1:X \ge x$$ where the point $x$ is covered by the rod of length $X$.  

*Question 2 concerns the event
$$\mathcal{E}_2:X \le x \le Y \text{ or } Y \le x \le X$$ where the point $x$ is covered by a rod with termini at $X$ and $Y$.  This event is shown by the gray portion of the figure (which extends, in principle, infinitely far to the right and infinitely far upwards).
The distribution of $X$ is often described by stating what probability is assigned to the event $X\le x.$  This is the distribution function (aka CDF) of $X,$ $$F_X(x)=\Pr(X\le x).$$  In these terms, question (1) is answered using basic laws of probability (namely, the probabilities of disjoint events add) as
$$\eqalign{
\Pr(\mathcal{E}_1) &= \Pr(X=x\text{ or } X \gt x) \\
&= \Pr(X=x) + 1 - \Pr(X\le x) \\
&= \Pr(X=x) + 1 - F_X(x).
}$$
The joint distribution of $(X,Y)$ can be described by stating what probability it assigns to infinite quadrants whose upper right corners are at a location $(x,y).$  This is the distribution function $$F_{XY}(x,y) = \Pr(X\le x\text{ and } Y \le y).$$
The figure shows we can express the probability of the gray region as equal to


*

*The probability of the entire region to the left of or at $X=x,$ equal to $F_X(x),$

*plus the probability of the entire region below or at $Y=x,$ equal to $F_Y(x),$

*minus the probability of the intersection of those regions (shown as the square with $(x,x)$ at its upper corner), equal to $F_{XY}(x,x).$
We see that the "interplay of events" is visually captured by this two-dimensional picture.  This is a general, powerful technique for reasoning about two random variables together.
Thus, the general answer to the second question is
$$\Pr(\mathcal{E}_2) = F_X(x)  +  F_Y(x) - F_{XY}(x,x).$$
The termini are said to be independent when $F_{XY}(x,y) = F_X(x)F_Y(y).$  In that case the general answer can be reduced to a formula in terms of $F_X$ and $F_Y$ alone by substitution in its last term.
