  Given a graphical degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $G(n, {\bf
d})$ denote a uniformly random graph on vertex set $[n]$ where vertex $ i$ has
degree $d_i$ for every $1\le i\le n$. We give upper and lower bounds on the
joint probability of an arbitrary set of edges in $G(n,{\bf d})$. These upper
and lower bounds are approximately what one would get in the configuration
model, and thus the analysis in the configuration model can be translated
directly to $G(n,{\bf d})$, without conditioning on that the configuration
model produces a simple graph. Many existing results of $G(n,{\bf d})$ in the
literature can be significantly improved with simpler proofs, by applying this
new probabilistic tool. One example we give is about the chromatic number of
$G(n,{\bf d})$.
  In another application, we use these joint probabilities to study the
connectivity of $G(n,{\bf d})$. When $\Delta^2=o(M)$ where $\Delta$ is the
maximum component of ${\bf d}$, we fully characterise the connectivity phase
transition of $G(n,{\bf d})$. We also give sufficient conditions for $G(n,{\bf
d})$ being connected when $\Delta$ is unrestricted.
