This problem is a by product of another problem. I would like to restate this problem as a sort of a puzzle.
Suppose we have a $l \times b$ grid. We select $k$ points on the grid randomly and independently assuming all points selected are distinct. That is $k$ distinct points on the grid are selected. The distance between two points is the euclidean distance between them. We are given a rope of length $n$. We want to connect all the points on the grid using this rope.
It can be noticed that this problem is similar to spanning tree problem. But we are allowed to cut the rope and tie the rope to the middle of another rope. For example below.
o Node1 | |
Node3 o-------------------------o Node2
The problem statement is, we just want to ensure all nodes are connected to each other.
It can be noted that for a given $k, n, l, b$ there exists probability that the nodes can be connected. For larger $l, b$ the probability is small and for smaller $l,b$ the probability that they can be connected is large.
Can we find a relation?