Product Rule

Topics: Probability

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(definition)

Let $A_1,\dots ,A_n\in \mathcal{F}$ (a sigma-algebra) such that $\mathbb{P}(A_1\cap \cdots \cap A_{n-1})>0$.

Then, the probability of the intersection of all these sets is:

$$\begin{array}{rl}\mathbb{P}(A_1\cap A_2\cap \cdots \cap A_n):=& \mathbb{P}(A_n\mid A_1\cap A_2\cap \cdots \cap A_{n-1})\\ & \mathbb{P}(A_{n-1}\mid A_1\cap \cdots \cap A_{n-2})\\ & \dots \\ & \mathbb{P}(A_3\mid A_1\cap A_2)\\ & \mathbb{P}(A_2\mid A_1)\\ & \mathbb{P}(A_1)\end{array}$$

(see conditional probability)