Exercise C.2.5

Prove that for any collection of events $A_1,A_2,\ldots,A_n$

$$ \Pr\{A_1 \cap A_2 \cap \cdots \cap A_n\} = \Pr\{A_1\} \cdot \Pr\{A_2 | A_1\} \cdot \Pr\{A_3 | A_1 \cap A_2\} \cdots \Pr\{A_n | A_1 \cap A_2 \cap \cdots \cap A_{n-1}\} $$

This is nice

$$ \begin{align} \Pr\{A_1 \cap A_2 \cap \cdots\} &= \Pr\{A_n | A_1 \cap \cdots \cap A_{n-1}\} \cdot \Pr\{A_1 \cap \cdots \cap A_{n-1}\} \\ &= \Pr\{A_n | A_1 \cap \cdots \cap A_{n-1}\} \cdot \Pr\{A_{n-1} | A_1 \cap \cdots \cap A_{n-2}\} \cdot \Pr\{A_1 \cap \cdots \cap A_{n-2}\} \\ &= \ldots \\ &= \Pr\{A_n | A_1 \cap \cdots \cap A_{n-1}\} \cdots \Pr\{A_3 | A_1 \cap A_2\} \cdot \Pr\{A_2 | A_1\} \cdot \Pr\{A_1\} \end{align} $$