Law of Total Probability¶

Theorem¶

Let $\{B_i\}$ be a finite or countable partition of the sample space into events. For every event $A$ ,

\mathbb{P}(A)=\sum_{i\colon\mathbb{P}(B_i)>0}\mathbb{P}(A\mid B_i)\mathbb{P}(B_i).

Null partition events contribute zero to $\mathbb{P}(A)$ and are omitted so that no undefined conditional probability is used.

Interpretation¶

Compute a probability by conditioning on each possible case and weighting by how likely that case is.

A rectangle split into three vertical strips B1, B2 and B3, with an oval event A crossing all three, so that A is cut into three pieces.

The diagram splits $A$ into the disjoint pieces $A\cap B_1$ $B_{1}$ , $A\cap B_2$ and $A\cap B_3$ $B_{3}$ . Their probabilities add to $\mathbb{P}(A)$ ; each positive-probability case contributes $\mathbb{P}(A\mid B_i)\mathbb{P}(B_i)$ $B_{i}) P (B_{i})$ .

Split the sample space into three cases and set $\mathbb{P}(A\mid B_i)$ in each: $\mathbb{P}(A)$ is the total shaded area, a weighted average of the three.

A partition of a set divides the sample space into disjoint blocks covering every outcome. Additivity of the probability function sums the contributions from $A$ within those blocks, and conditional probabilities express each contribution as a block probability times a within-block probability.

Key Results¶

$\displaystyle \mathbb{P}(A)=\sum_i\mathbb{P}(A\cap B_i)$ for a countable partition.
Only terms with $\mathbb{P}(B_i)>0$ need conditional probabilities; zero-probability cases contribute zero.

Examples¶

If factories supply $60\%$ and $40\%$ of products with defect rates $0.01$ and $0.03$ , the defect probability is $0.6(0.01)+0.4(0.03)=0.018$ $0.018$ .
For a fair die, conditioning on odd or even gives $\mathbb{P}(X=6)=0(1/2)+(1/3)(1/2)=1/6$ $1/6$ .

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 2
Nicky van Foreest · Probability Distributions · Week 3

Aliases: LOTP, total probability theorem, total probability formula, theorem of total probability, rule of total probability, law of total probabilities Symbols: intersection, sum

Last updated: 17 October 2026 21:53