TITLE: Edge probability for connected Erdős–Rényi model
QUESTION [7 upvotes]: Consider the Erdős–Rényi model $G_{n,p}$ with corresponding probability measure $\mathbb{P}_{n,p}$. For any two vertices $x,y$, $\mathbb{P}_{n,p}[E_{x,y}]=p$, where $E_{x,y}$ is the event that there exists an edge between $x$ and $y$.
I need to estimate (especially bound from above) the following probability for two fixed vertices $x,y$ and $G \in G_{n,p}$:
$\begin{equation} 
\mathbb{P}_{n,p}[E_{x,y}|G \text{ is connected}]
\end{equation}$
Supplement: $p=\frac{c}{n}$ for some constant $c >1$ (I forgot this in my first version).

REPLY [2 votes]: I agree strongly with Peter in the comments: if you're interested in $p \gg \frac{\log n}{n}$, then don't do any hard work, just show the answer is essentially $p$ since the graph is essentially always connected.
In particular, let $E$ be the event that the edge exists and $C$ the event that the graph is connected. By the law of total probability,
\begin{align*}
  P(E|C)P(C) + P(E,\lnot C) &= p
\end{align*}
so
\begin{align*}
  P(E|C) &= \frac{p - P(E,\lnot C)}{P(C)}  \\
         &\leq \frac{p}{1-o(1)}
\end{align*}
where a lot is known about bounds on the $o(1)$ (the probability the graph is disconnected).
(By the way, naturally, we immediately have $P(E|C) \geq p$. To prove it, $P(E|C) = p P(C|E)/P(C)$, and $P(C|E) \geq P(C)$ as guaranteed existence of $(x,y)$ can only raise the chance of connectedness.)