The total probability theorem states the following:

Let $A\_1$,...,$A\_n$ be a partition of $\Omega$. For any event B,

$$Pr(B)=\sum\_{i=1}^n Pr(A\_i)Pr(B|A\_i)$$

We know that $Pr(B|A\_i)= \frac{Pr(B\cap A\_i)}{Pr(A\_i)}$, therefore the $Pr(A\_i)Pr(B|A\_i)$ can simplify to $Pr(B\cap A\_i)$.

Couldn't the above theorem then be directly expressed as:

$$Pr(B)=\sum\_{i=1}^n Pr(A\_i\cap B)$$

If yes, then why use the above representation?