Section 5.1

The Law of Total Probability

Conditioning on an intermediate random outcome: balls transferred between urns, a random number of trials, a randomly chosen die, sequential draws.

Toolkit

The Law of Total Probability turns a hard unconditional question into a weighted average over a convenient partition $\{B_i\}$ of the sample space:

\mathbb{P}(A)=\sum_i \mathbb{P}(A\mid B_i)\,\mathbb{P}(B_i),\qquad \mathbb{E}[X]=\sum_i \mathbb{E}[X\mid B_i]\,\mathbb{P}(B_i).

Choose $B_i$ to be whatever intermediate randomness, once revealed, makes the remaining computation trivial: the origin urn of a transferred ball, the number of trials in a random-length experiment, the realized value of a random parameter, the first step of a chain, or the stage at which an item is removed. For a continuous parameter $\Theta$ with density $f$ ,

\mathbb{P}(A)=\int \mathbb{P}(A\mid \Theta=\theta)f(\theta)\,d\theta.

Symmetry across the partition (e.g. exchangeability, or a flipped-parity pairing) often collapses the sum to a strikingly simple closed form without evaluating each term.

Problems