Extending the Law of Total Probability

You've encountered the Law of Total Probability for a two-event partition. We now extend it to any number of events.

A partition of the sample space $S$ is a collection of events $B_1, B_2, \ldots, B_n$ that are pairwise disjoint and whose union is $S$. Every outcome of the experiment belongs to exactly one of the $B_i$.

The extended Law of Total Probability states that for any event $A$ and any partition $B_1, \ldots, B_n$ of $S$,

$$P(A) = \sum_{i=1}^{n} P(A \mid B_i)\, P(B_i) = P(A \mid B_1) P(B_1) + P(A \mid B_2) P(B_2) + \cdots + P(A \mid B_n) P(B_n).$$

For example, suppose $B_1, B_2, B_3$ partition $S$ with

$$P(B_1) = 0.5,\quad P(B_2) = 0.3,\quad P(B_3) = 0.2,$$

and the conditional probabilities of an event $A$ are

$$P(A \mid B_1) = 0.4,\quad P(A \mid B_2) = 0.2,\quad P(A \mid B_3) = 0.1.$$

Then

$$P(A) = 0.5 \cdot 0.4 + 0.3 \cdot 0.2 + 0.2 \cdot 0.1 = 0.20 + 0.06 + 0.02 = 0.28.$$

Notice that the weights $P(B_i)$ must sum to $1$, since the $B_i$ partition $S$.