Alternative formulation of the Law of Total Probability Usually the following formulation of the Law of Total Probability is used:


  If events $\{H_i\}$ with positive probabilities form the partition of a sample space then for any event $A$ on the same probability space we have $P(A) = \sum_{i} P(A|H_i)P(H_i)$


Do we rigorously need the partition here? Can we use the following alternative formulation instead?


  If pairwise disjoint events $\{H_i\}$ with positive probabilities form the cover of the event $A$ (i.e. $A \subseteq \bigcup_i H_i$) then we have $P(A) = \sum_{i} P(A|H_i)P(H_i)$.
Hint: What is $\Pr ( A \vert H_i )$ when $A \cap H_i = \emptyset$?

This should tell you the connection between the two statements.
Once you have pairwise disjoint non-empty sets $\{ H_i \}$ that cover $A$, that gives you a partition --i.e., the class $\{ H_i \cap A \}$ partitions $A$.  So your second statement is not really an alternative result at all - it is just a way of stating the same result.