Law of total probability

Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $B_1, B_2, \dots$ be a finite or countable family of pairwise disjoint events with $\bigcup_i B_i = \Omega$ and $P(B_i) > 0$ for each $i$. Then for each event $A$, $$P(A) = \sum_{i} P(A \mid B_i) \, P(B_i).$$

Remarks

The events $B_i$ form a partition of $\Omega$ into events of positive probability. The proof uses $A = \bigcup_i (A \cap B_i)$, countable additivity and the multiplication rule. The case of two events $B$ and $B^c$ is the law of total probability for an event and its complement. Example: urn $1$ has $2$ red balls of $5$ and urn $2$ has $3$ red balls of $4$. Choose an urn with a fair coin and draw one ball. The ball is red with probability $\frac12 \cdot \frac25 + \frac12 \cdot \frac34 = \frac{23}{40}$.

Explanations

L1 human

Split the world into cases that do not overlap. The chance of $A$ is the average of its chances in the cases, weighted by the chances of the cases.