Consider a set of events $A_1, \dots, A_n$.

By definition we have

$$\mathbb{P}[A_1 \cap (A_2 \cap \dots \cap A_n)] = \mathbb{P} [(A_2 \cap \dots \cap A_n) | A_1] \mathbb{P}(A_1).$$

Applying the definition again,

$$ \mathbb{P} [A_2 \cap (A_3 \cap \dots \cap A_n) | A_1] = \mathbb{P} [ (A_3 \cap \dots \cap A_n) | A_1, A_2] P(A_2|A_1),$$

which together with the first equality implies

$$\mathbb{P}[A_1 \cap (A_2 \cap \dots \cap A_n)] = \mathbb{P} [ (A_3 \cap \dots \cap A_n) | A_1, A_2] \mathbb{P}(A_2|A_1) \mathbb{P}(A_1).$$

Continuing the argument recursively yields the following often used decomposition:

$$\mathbb{P}[A_1 \cap A_2 \cap \dots \cap A_n] = \mathbb{P}(A_1) \mathbb{P}(A_2|A_1) \dots \mathbb{P}(A_{n}|A_1, A_2, \dots, A_{n-1}) .$$

My question is, is there a standard terminology to refer to this decomposition (say as standard as the "Law of Total Probability" terminology)?

  • 1
    $\begingroup$ Closely related to the prediction-error decomposition $p(X_n,X_{n-1},...,X_2,X_1) =$ $p(X_n|X_{n-1},...,X_2,X_1) \cdot p(X_{n-1}|X_{n-2}...,X_2,X_1)\cdot ...\cdot p(X_2|X_1) \cdot p(X_1)$ $\endgroup$
    – Glen_b
    May 24, 2019 at 7:52

1 Answer 1


That is repetitive implementation of the definition of conditional probability as you describe, and is referred as Chain Rule. More generally, it can be written as $$P\left(\bigcap_{k=1}^n A_k\right)=\prod_{k=1}^n P\left(A_k\bigg\vert\bigcap_{m=1}^{k-1}A_j\right)$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.