Uniform distribution over the square region.

Question

Given: $(X,Y)$ be uniformly distributed over the square $R = \{(x; y) : 0 < x < 1; 0 < y < 1 \}$ . Let $U = min(X; Y )$ and $V = max(X;Y)$.

My attempt to show that: $U$ and $V$ are not independent by picking two subsets $\subset [0,1]$:

But how do I formulate the expression in words:

$$ P(U \in A, V \in B)$$ where $A,B \subset [0,1]$.

I already find the product of $P(U \in A) \times P(V \in B)$.

Answer 1

Let $A=(\frac 1 2 ,1)$ and $B=(0,\frac 1 2)$. Then $P(U \in A,V \in B)=0$ because $V \geq U$. But $P(U \in A)=\frac 1 4 =P(V \in B)$.