Even before Odd

(Product Formula)

Let $A_1, A_2, \dots, A_n$ be events such that:
$\mathbb{P}\left(\bigcap_{k=1}^n A_k\right) > 0$
Show that:
$\mathbb{P}\left(\bigcap_{k=1}^n A_k\right) = \prod_{k=1}^n \mathbb{P}\left(A_k \,\middle|\, \bigcap_{j=1}^{k-1} A_j\right)$
(Even before Odd)

A fair 6-sided die is rolled repeatedly until all six faces $\{1,2,3,4,5,6\}$ have been observed at least once.

What is the probability of observing no odd face before having observed all even faces?

Answer