Total Probability Theorem: Formula, Tree Diagrams, Examples

The theorem of total probability computes the probability of an event A that can happen through several different, mutually exclusive routes. Each term is one route: the chance of taking that route times the chance A happens along it. Adding the routes gives the total chance of A.

You cannot compute P(A) directly, but you can compute it within each group and weight by group sizes. The theorem is also the denominator of Bayes theorem, and it is the natural way to read a tree diagram: multiply along each branch, then add across branches.

Key Concepts

• Partition: events E1 … En that are pairwise mutually exclusive and whose union is the whole sample space; exactly one occurs.
• Prior probabilities P(Ei): the chance each route is taken, before knowing anything about A; they must sum to 1.
• Likelihoods P(A | Ei): the chance A happens given that route Ei was taken.
• Weighted sum: each likelihood is weighted by its route's probability, so rare routes contribute little even with high likelihood.
• Tree diagram: first-level branches show the partition with P(Ei), second level shows P(A | Ei); multiplying down each path and adding gives P(A).

Formulas

For a partition E1 … En: P(A) = Σ P(Ei) × P(A | Ei), summed over i = 1 to n. Two-case version: P(A) = P(E) × P(A | E) + P(E′) × P(A | E′). The priors must satisfy P(E1) + … + P(En) = 1. Each term P(Ei) × P(A | Ei) equals P(A ∩ Ei) by the multiplication theorem, so the formula is the addition theorem applied to the disjoint pieces A ∩ Ei.

Worked Example

Urn I contains 3 red and 2 white balls; Urn II contains 2 red and 4 white balls. A fair coin chooses the urn, then one ball is drawn. Find P(red). The partition is E1 = Urn I chosen, E2 = Urn II chosen, with P(E1) = P(E2) = 1/2. P(red | E1) = 3/5 and P(red | E2) = 2/6 = 1/3. By total probability, P(red) = (1/2) × (3/5) + (1/2) × (1/3) = 3/10 + 1/6 = 9/30 + 5/30 = 14/30 = 7/15, about 0.467. As a tree: branch Urn I (1/2) then red (3/5) gives 3/10; branch Urn II (1/2) then red (1/3) gives 1/6; adding the two red paths gives 7/15.

Common Misconceptions

• Averaging the likelihoods: P(A) is not the simple average of the P(A | Ei); each is weighted by P(Ei). A rarely chosen urn barely affects the total.
• Using overlapping routes: If the Ei overlap, the pieces A ∩ Ei overlap too and adding them double-counts. The partition must be mutually exclusive.
• Forgetting a route: If the Ei are not exhaustive, probability mass is missing and the total comes out too small; the priors must sum to 1.

Real-World Applications

Insurance actuaries blend claim rates across driver categories. Anywhere a population splits into groups with different behaviour, the overall rate is a total-probability sum.

The total probability theorem computes P(A) = Σ P(Ei) × P(A | Ei) over a partition, weighting each route's likelihood by its probability. It is read directly off a tree diagram by multiplying along branches and adding across them. It handles every problem where an event can occur through several exclusive channels, and it supplies the denominator for Bayes theorem.