I have random variables $X_1,\ldots,X_n \in \{0,1\}$ which are dependent and $\mathbb{P}[X_i=1 | X_1=x_1, \ldots, X_{i-1}=x_{i-1}, X_{i+1}=x_{i+1}, \ldots X_{n}=x_n] \geq p$ for any $x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n \in \{0,1\}$.
I also have random variables $Z_1,\ldots,Z_n \in \{0,1\}$ which are independent and $\mathbb{P}[Z_i=1]=p$.
What is a simple and yet rigorous way to argue that for any integer $k\geq0$, $\mathbb{P}[X_1 + \ldots + X_n > k] \geq \mathbb{P}[Z_1 + \ldots + Z_n > k].$
(I have a very ugly way right now, but it seems that there should be a one-line argument.)