If $m,n$ are co-prime , we know that $\phi(mn)=\phi(m)\phi(n)$.
I want to prove it using probability.
Probability that a selected number less than or equal to $mn$ is co-prime to $mn$ = $\dfrac{\phi(mn)}{mn}$
Probability that a selected number less than or equal to $m$ is co-prime to $m$ = $\dfrac{\phi(m)}{m}$
Probability that a selected number less than or equal to $n$ is co-prime to $n$ = $\dfrac{\phi(n)}{n}$
How to define events and sample space and then how to prove the last two events are independent in-order to prove desired one?