Question

Let x and y be independent each uniformly distributed on {1,2,...,n} find p(x < y )

Answer 1

If we draw the contingency table of x (vertical) against y (horiz.), we have a square.
For n=4, we have   (legend:  < : x<y   = : x=y   > : x>y

y      1  2  3  4
x
1      = <  <  <
2      > =  <  < 
3      >  >  = <
4      >  >  >  =

We see that there are n(n-1)/2 cases of x<y out of n^2.
Therefore, 
p(x<y)=n(n-1)/(2n^2)=(n-1)/(2n)

However, if the sample space is continuous, it will be simply p(x<y)=1/2.