TITLE: X,Y ~ Unif(0,1) not necessarily independent, can P(X+Y>1)>1/2?
QUESTION [8 upvotes]: My set up is the following:
$X,Y \sim \text{Unif}(0,1)$ but their joint distribution is not constrained. My question is whether there exists a joint dependence between them (that preserves the marginals) such that $\operatorname{Prob}(X+Y>1)>1/2$.
I can show it is = 1/2 for independence (via integrating the joint PDF), but am wondering whether there is a simple argument or counterexample either way for the cases where the joint distribution is not constrained. I have tried conditioning on one of the variables but couldn't make progress. All the simiulation evidence I have suggests it is = 1/2 for a variety of dependencies.
Thanks!

REPLY [4 votes]: I think the idea of @Jean Marie of providing a visual solution is great. However, I think a closest ``discrete version'' of the answer can rather be seen on the 4x4 grind, which I think would clarify things.
In fact you only need to fill the first boxes over the diagonal and the bottom-left corner. Here is a view for a 4x4 grid :

And here it is for the 5x5 grid :

Clearly, both random variables are uniform (we have 1 filled square per line, and one per column). Moreover, the only square filled such that $X+Y \le 1$ represents $1/n$ of the colored area, when we consider a $n \times n$ grid. Hence we can find a decomposition arbitrarely close to $1$ by playing with $n$.