Evaluating the expected value of negatively correlated random variables A polynomial random process satisfying the following properties converts a fractional point $(x_1, x_2, \ldots, x_n) \in \mathcal{P}$, $(x_i \in [0,1])$ to a random integer point $(X_1, X_2, \ldots, X_n) \in \mathbb{Z}(\mathcal{P})$, $(X_i \in \{0,1\})$:


   $\mathbb{E}[X_i]=x_i$, for all $i \in [n]$.

  For any $S \subseteq [n]$, $\mathbb{E}[\prod_{i\in S} X_i] \leq \prod_{i \in S} x_i$ and $\mathbb{E}[\prod_{i\in S} (1-X_i)] \leq \prod_{i \in S} (1-x_i)$ (negative correlation).


An example is the dependent randomized rounding by Chekuri, Vondrak, and Zenklusen (http://arxiv.org/pdf/0909.4348v2.pdf). 

We focus on the dependent randomized rounding. Given a point $x \in \mathcal{P}$, where $\mathcal{P}$ is a matroid polytope, let us assume that $X$ is the outcome of the random process (the dependent randomized rounding). Can we evaluate the value of $Pr\Big(\displaystyle \bigwedge_{i \in S} X_i=1 \Big)$  for some $S \subseteq [n]$? Shall we use sampling for this purpose?

That would be great if you can comment on this.
I think that in principle it would take exponential time to compute this probability exactly. Hence sampling would be the only option here, although you would have to settle for additive and multiplicative approximation here.
Can I ask you about the problem in which you need this value to be computed?