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Is there a name for the following identity? $$ \begin{align*} & \Pr\left(\bigwedge_{i=1}^n A_i \mid B \right)\\ &= \Pr\left(A_1 \mid B \right) \cdot \Pr(A_2 \mid A_1 \wedge B) \cdot \Pr(A_3 \mid A_1 \wedge A_2 \wedge B) \cdots \Pr\left(A_n \mid \bigwedge_{i = 1}^{n-1} A_i \wedge B \right) \end{align*} $$

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up vote 2 down vote accepted

This looks like the product rule or chain rule or multiplication rule for the probability of the intersection of $n$ events: $$P(A_1A_2\cdots A_n) = P(A_1)P(A_2|A_1)P(A_3|A_1A_2)\cdots P(A_n|A_1A_2\cdots A_{n-1})$$
except that everything is conditioned on the event $B$. See, for example, S. Ross A First Course in Probability, Eighth edition, page 63.

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