Total Probability Theorem — Definition, Formula & Examples

The Total Probability Theorem lets you find the probability of an event by breaking the sample space into separate, non-overlapping scenarios and adding up the probability of the event occurring in each scenario.

If $B_1, B_2, \ldots, B_n$ form a partition of the sample space (mutually exclusive and exhaustive events with $P(B_i) > 0$ ), then for any event $A$ : $P(A) = \sum_{i=1}^{n} P(A \mid B_i)\,P(B_i)$ .

Key Formula

P(A) = \sum_{i=1}^{n} P(A \mid B_i)\,P(B_i)

Where:

$A$ = The event whose total probability you want to find
$B_i$ = The i-th event in the partition of the sample space
$P(A \mid B_i)$ = Conditional probability of A given scenario B_i
$n$ = Number of mutually exclusive scenarios in the partition

How It Works

Start by identifying a set of mutually exclusive scenarios that cover every possibility — these form your partition. For each scenario

B_i

, find the conditional probability

P(A \mid B_i)

and the probability

P(B_i)

of that scenario itself. Multiply these two values together for each scenario, then add all the products. The result is the overall probability

P(A)

. This approach is especially powerful when

P(A)

is hard to compute directly but easy to compute within each scenario.

Worked Example

Problem: A factory has two machines. Machine 1 produces 60% of all items and has a 3% defect rate. Machine 2 produces the remaining 40% and has a 5% defect rate. What is the probability that a randomly chosen item is defective?

Identify the partition: The two machines form a partition:

B_1

= made by Machine 1,

B_2

= made by Machine 2.

P(B_1) = 0.60, \quad P(B_2) = 0.40

List conditional probabilities: The defect rates give us the conditional probabilities of event

A

(item is defective).

P(A \mid B_1) = 0.03, \quad P(A \mid B_2) = 0.05

Apply the Total Probability Theorem: Multiply each conditional probability by the probability of its scenario, then add.

P(A) = (0.03)(0.60) + (0.05)(0.40) = 0.018 + 0.020 = 0.038

Answer: The probability that a randomly chosen item is defective is

0.038

, or

3.8\%

Why It Matters

The Total Probability Theorem is a key stepping stone to Bayes' Theorem — you almost always need it to compute the denominator in Bayes' formula. It appears in AP Statistics, college probability courses, and real-world fields like medical testing, quality control, and risk analysis where outcomes depend on multiple underlying conditions.

Common Mistakes

Mistake: Using scenarios that are not mutually exclusive or do not cover the entire sample space.

Correction: Your scenarios

B_1, B_2, \ldots, B_n

must form a true partition: they cannot overlap, and together they must account for every possible outcome. Verify that

P(B_1) + P(B_2) + \cdots + P(B_n) = 1