Theorem of Total Probability

13.5.2 Theorem of Total Probability

Understanding the Need for This Theorem

When we want to find the probability of an event $A$, we often know how $A$ behaves under different conditions or scenarios. For example, the probability that a construction job finishes on time depends on whether there is a strike or not. The Theorem of Total Probability gives us a systematic way to combine these conditional probabilities into one overall probability.

The key idea is to break the sample space into mutually exclusive and exhaustive pieces — a partition — and then express $P(A)$ as a weighted average of the conditional probabilities $P(A|E_i)$, where the weights are the probabilities $P(E_i)$ of the pieces themselves.

Partition of a Sample Space

Before stating the theorem, recall what a partition means. A collection of events $\{E_1, E_2, \dots, E_n\}$ is called a partition of the sample space $S$ if:

• $E_i \cap E_j = \phi$ for all $i \neq j$ (the events are pairwise disjoint — no two overlap)
• $E_1 \cup E_2 \cup \dots \cup E_n = S$ (their union covers the entire sample space)
• $P(E_i) > 0$ for each $i$ (each event has a non-zero probability of occurring)

In everyday language, a partition divides the sample space into non-overlapping pieces that together account for every possible outcome. Exactly one of the $E_i$ must occur.

Statement of the Theorem of Total Probability

Proof of the Theorem

The Special Case of Two Events

When the partition consists of just two events $E$ and $E'$ (where $E'$ is the complement of $E$), the theorem simplifies to:

This two-event form is extremely common in applications — for example, when an outcome depends simply on whether or not a particular event occurs, splitting the sample space into that event and its complement.

Connection to Bayes' Theorem …