Suppose we construct an undirected graph called Node Map, consisting of n nodes and n edges. The graph contains at least 9 nodes, and the number of nodes is always an integer multiple of three. The graph is constructed so that it can be represented as three unconnected cycles that each have the same number of nodes and edges. If our value for n was 12 our Node Map may look like this:

Figure 1 A set of three unique nodes from the Node Map graph, [nα, nβ,nγ], is selected entirely at random from the set of all nodes in Node Map that are not currently connected by an edge. This means that once a node is randomly selected, any node it is currently connected to by an edge is removed from the potential pool of nodes that can be chosen in any subsequent random selections. As an example, if the first node randomly selected, nα, was the orange shaded node in the diagram above, neither of the nodes shaded blue could be randomly selected as nβ or nγ.

With the three nodes selected, a new graph called Node Updated is drawn. In Node Updated, two new edges are drawn, one connecting nα and nβ, and one connecting nβ and nγ . What is the probability that Node Updated is a connected graph?