Probability tree

A probability tree is a branching diagram mapping sequential events, with conditional probabilities on each branch and outcomes found by multiplying along a path.

ByHoang TruongUpdated July 11, 2026

See it move

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Machine 1 makes 60% of parts with a 5% defect rate, so its path probability is 0.60 × 0.05, or 0.03. Machine 2 makes 40% with a 10% defect rate, giving 0.40 × 0.10, or 0.04. Adding both paths, the overall probability a part is defective is 0.03 + 0.04, or 7%.

Where it fits

TopicProbability & DistributionsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

P(A \cap B) = P(A)\,P(B\mid A)

Variables

$A$: First-stage event (first branch)
$B$: Second-stage event (next branch)
$P (B ∣ A)$: Conditional probability of B given A

Gives the probability of a single path through a probability tree by multiplying along its branches.

LaTeX

P(X) = \sum_i P(A_i)\,P(X\mid A_i)

Variables

$A_{i}$: The i-th first-stage branch
$X$: The outcome of interest
$P (X ∣ A_{i})$: Conditional probability of X given branch A_i

Gives the overall probability of an outcome by summing across every first-stage branch that can lead to it.

Check yourself

PracticeCORE

A factory buys parts from two suppliers: Supplier A provides 70% of parts with a 2% defect rate, and Supplier B provides the remaining 30% with a 6% defect rate. What is the probability that a randomly chosen part is defective?

Select an answer to check your understanding.