Let $$X\sim\mathbb{N}([0,1,0]^{\rm T}, \mathrm{\Sigma})$$ and $$Y\sim\mathbb{N}([1,0,0]^{\rm T}, \mathrm{\Sigma}).$$ For some real constant $c$ what can be said about $$\mathbb{P}[\cap_{i=1}^3|X_i|\leq c]$$ and $$\mathbb{P}[\cap_{i=1}^{3}|Y_i|\leq c]?$$ Which probability is more?
I have some numerical results that shows these two probabilities are not same in general (for any $\mathrm{\Sigma}$). Is there any stochastic ordering result to compare these probabilities?