<s>[INST] Consider the unit square in the coordinate plane with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$, and suppose that $S$ is a sample of 1000 points chosen uniformly at random from this square.  Let $X$ denote the leftmost point in $S$, and $Y$ the topmost point.
It is given that $P(X<0.5)=P(Y<0.5)=\frac{1}{2}.$

Also given that the events $X<0.5$ and $Y<0.5$ are strongly positively correlated.

Consider a line with equation $y=b$ that divides the square into two equal regions. Without the loss of generality, assume $X \leq Y$. Since the points are chosen uniformly at random, given the equation $y=b$, it follows that exactly half of $S$ will lie on or below $y=b$ and exactly half will lie on or above $y=b$.

Now to this co-ordinate plane, superimpose the following lattice: a grid of vertical and horizontal lines each of length 1 that intersect the square in the center. This means that we will have four sub-regions (left, right, top, bottom) of area 0.25.

By observing this grid, I can see that half of S will lie on or below y = b and the other half on or above y = b.

What is the least value of b such that s(X < 0.5, Y < b) ≥ 1/2 if the probability of both X and Y being less than 0.5 cannot be less than 50%?[/INST]