A graph consists of a finite set of vertices, and edges between some (unordered) pairs of these vertices. The graphs in this question have no loops (an edge between a vertex and itself) or multiple edges (more than one edge between the same pair of vertices). A vertex $v$ in a graph is said to be a global vertex if there is an edge between $v$ and every other vertex in the graph.
A graph $G$ is constructed as follows: First, its vertex set is assigned to be the set $\left\{v_{1}, v_{2}, \dots, v_{10}\right\}$. Next, between each pair of distinct vertices $v_{i}, v_{j}$, an edge is added with probability $\frac{1}{2}$.
- What is the probability that vertex $v_{1}$ is a global vertex in graph $G$?
- What is the expected number of global vertices in graph $G$?
