Law of Total Probability

Stitching together cases that partition the sample space.

When you can't compute $P(A)$ directly, condition on something you can. If $\{B_1, B_2, \dots, B_n\}$ is a partition of the sample space — pairwise disjoint and jointly covering all of $\Omega$ — then for any event $A$ :

P(A) = \sum_{i=1}^{n} P(A \mid B_i)\, P(B_i)

The pieces are usually easier than the whole. To compute $P(\text{rain tomorrow})$ , condition on whether the wind shifts overnight; to compute $P(\text{system fails})$ , condition on which subsystem went down.

The same idea works for expectations:

E[X] = \sum_{i} E[X \mid B_i]\, P(B_i)

This is the engine behind recursive expectation problems. The classic example: the expected number of fair-coin flips until two heads in a row. Let $E_0$ be the expectation from a clean state and $E_1$ the expectation after seeing one head. Conditioning on the next flip gives a small linear system you can solve in seconds.