I can't seem to get my head around a following conclusion:
Def: H is a finite set of hashing functions from U to [m] (set of size m). We'll call H universal if: $\forall x,y; x\ne y $ there exists $\frac{|H|}m$ functions $h\in H$ such that $h(x)=h(y)$.
Now, the conclusion should be that for random chosen $h\in H $ and $\forall x,y; x\ne y $ chance of collision between x,y is $\frac{1}m$.
Could anyone explain it to me, please?
My solution:
$\forall x,y;x\ne y$ is the chance that we pick $h\in H$ that causes a collision following fraction: $\frac{\text{Functions that cause collision}}{\text{All functions}}=\frac{\frac{|H|}m}{|H|}=\frac{1}m$
Is it correct?