Joint Distributions of Uniforms

Let \(X\) and \(Y\) be independent, each uniformly distributed on \(\{1,2,\dots,n\}\). Find:

1. \(\mathbf{P}( X=Y)\)
2. \(\mathbf{P}( X < Y)\)
3. \(\mathbf{P}( X>Y)\)
4. \(\mathbf{P}( \text{max}(X,Y)=k )\) for \(k=1,\dots,n\)
5. \(\mathbf{P}( \text{min}(X,Y)=k )\) for \(k=1,\dots,n\)
6. \(\mathbf{P}( X+Y=k )\) for \(k=2,\dots,2n\)