Law of Total Probability

P(A) = _i P(A | B_i) P(B_i) — condition and weight.

The idea

Theorem (Law of total probability).

Suppose the events $B_1, \ldots, B_n$ partition the sample space, meaning exactly one of them occurs. Then for any event $A$, $\mathbb{P}(A) = \mathbb{P}(A \mid B_1)\,\mathbb{P}(B_1) + \cdots + \mathbb{P}(A \mid B_n)\,\mathbb{P}(B_n).$

The formula computes a probability that is hard to see directly from probabilities that are easy to see once the case is known. We split the situation into the cases $B_1, \ldots, B_n$, find the chance of $A$ inside each case, and recombine the pieces.

Each conditional probability enters with a weight. $\mathbb{P}(A \mid B_i)$ is the chance of $A$ given that $B_i$ occurs, so it should count only as often as $B_i$ itself occurs, which is a $\mathbb{P}(B_i)$ fraction of the time. Adding the conditional probabilities without their weights counts every case as though it were certain, and can produce a total larger than $1$.

When the cases are equally likely, the weights are all equal and the formula is an ordinary average.

Ways to work on it

• Walkthrough. Compute an overall probability by conditioning on which case occurred.
• Practice. Equal-weight partition: just average two probabilities.
• Hardest. Compute P(A) from a partition in a single step.

Not sure where to start? Take the ten-question placement test.