What is the probability that the distance from L to 0 is smaller than the distance from M to 0? 
Two points L and M belongs to the interval [0, 1] are chosen at
random. Thus, the interval is divided into 3 smaller sections.
What is the probability that the distance from L to 0 is smaller than
the distance from M to 0?

My attempted solution.
The condition stands if we have,
$0L < 0M$
$\Rightarrow M-L > 0$
$\Rightarrow M > L$
Since, the plotted line passes through the diagonal of the unit square, the area is $\frac{1}{2}$.
So, the probability is, $\frac{1/2}{1}=\frac{1}{2}$.

Is this solution correct? Why or why not?
 A: You should always apply a reasonability check to your answers.  That is, ask yourself:

Is it reasonable that the probability is zero?

In your case, you are modeling throwing two darts at a line segment $[0, 1]$, and want to know the probability that the first dart is closer to zero than the second dart.  A probability of zero would mean that the situation is impossible, which it is clearly not!
The correct answer is $\frac{1}{2}$, which you can arrive at in multiple ways.  One simple way is to observe that the roles of $L$ and $M$ in the problem are interchangeable, you know exactly the same things about both.  Therefore the answer cannot depend on the labeling of one varaible as $L$ and the other as $M$, so the following probabilities must be equal
$$P(L > M) = P(M > L)$$
Since these probabilities must sum to $1$, it follows that they must both be $\frac{1}{2}$(*).
(*) I've snuck in a small assumption that $P(L = M) = 0$.  Do you see where?
A: No the answer should be 1/2.  It is the probability that one uniform random variable is larger than another independent one.
