Suppose we wish to construct a graph in the following manner: Denote the vertices of the graph as $1,2,\dots,n$. For every pair $\{i,j\}$, we flip a fair coin. If it comes up tails, $\{i,j\}$ is an edge of the graph; if it comes up heads, $\{i,j\}$ is not an edge of the graph. Please answer the following:
a. What is the probability that the graph we end up with is a complete graph (i.e., $K_n$) ?
b. Let $D_1$ denote the degree of vertex $1$, What is $\Bbb E[D_1]$, the expected degree of vertex $1$?
c. Notice that the expected degrees of vertices $2,\dots,n$ should also be the same as that of vertex $1$. Given this observation, and the handshaking lemma, what is the expected number of edges in the graph?
I have no idea about this question,