# Total Probability Theorem / Partition

In the Total Probability Theorem we assume that the sample space is partitioned into subsets. If we consider $B$ to be the sample space and $A_1$, $A_2$ to be the partition then the theorem says:

$$\mathbb{P}(B)=\mathbb{P}(A_1) \mathbb{P}(B\mid A_1)+\mathbb{P}(A_2) \mathbb{P}(B\mid A_2)$$

My confusion is that if $A_1$ and $A_2$ are subsets of $B$, should not $\mathbb{P}(B\mid A_1)=1$?

That's correct: $\mathbb{P}(B\mid A_1)=\mathbb{P}(B\mid A_2)=1$, assuming that $A_1$ and $A_2$ have positive probability. Therefore the total probability theorem in this case reduces to $$\mathbb{P}(B)=\mathbb{P}(A_1)+\mathbb{P}(A_2)$$ which must hold if $A_1$ and $A_2$ form a partition of $B$.
The more interesting applications of the total probability theorem come when you have a partition $\{A_n\}$ of the sample space $\Omega$ and some other event $B$ you want to find the probability of.