What is the probability that there is an edge in an undirected random graph having m vertices?[assume prob of edge between 2 vertices - 1/2]
how to think in this type of problem.
probability that an edge exists between 2 vertices = 1/2
$(1/2)^{m-1}$ is probability of m length path.
Am I misreading this? By definition every edge exists between precisely two vertices, except for loops which exist on one. Between fewer than 1 vertices you havent got an edge at all. The point is, every edge has two endpoints. A single edge cannot connect more than two vertices. Thus $m$ can only be 1 or 2. For all other values of $m$ the probability must necessarily be 0 because the scenario is impossible.
Since every complete graph has $\binom{m}{2}$ two-vertex edges, and potentially $m$ one-vertex edges (loops), then in a completely random graph there are $\binom{m}{2} + m $ total edges. These are the only possible undirected edges in a simple graph. If you want generalize then there is an infinitude of possible edges between any two vertices.
So $\binom{m}{2} + m $ is the possible number of edges. Or simply $\binom{m}{2}$, if there are no loops. This is the denominator of any probability fraction you come up with.