Are bins with more black than white balls negatively 1-correlated?
A good reference is Concentration of Measure for the Analysis of Randomized Algorithms, by Dubhashi and Panconesi.  This is not a standalone proof but shows how you could use the results therein to prove your desired inequality.

We say a set of random variables $X_i, i \in [n]$ are negatively associated if for all disjoint subsets $I,J \subset [n]$, and all nondecreasing functions $f$ and $g$,
$$
\mathbb{E}[f(X_i, i \in I)g(X_j, j \in J)] \leq \mathbb{E}[f(X_i, i \in I)]\mathbb{E}[g(X_j, j \in J)].
$$

By the proof of proposition 3.1 of D&amp;P, the variables $\{B_1, \dots, B_n\}$ and $\{W_1, \dots, W_n\}$ are negatively associated.  By the disjoint monotone aggregation property of negatively associated variables, (p.35 of D&amp;P), $-W_1, \dots, -W_n$ are also negatively associated, and the random variables $B_i-W_j$, over all choices of $i,j$ are as well.  In particular, the set of variables $B_i - W_i$, for any subset $I \subset [n]$ are negatively associated.

Then, per exercise 3.1, or just by applying the above inequality recursively with $f=g=1$,
$$
\mathbb{E}[ \prod_{i \in I}M_i ] \leq \prod_{i\in I}\mathbb{E}[ M_i],
$$
from which your inequality follows.