multiplication rule for conditional probability

The probability of two events occurring simultaneously can be expressed by rewriting the definition of conditional probability as

\begin{aligned} P (A \cap B) & = P (A ∣ B) \cdot P (B) \\ = P (B ∣ A) \cdot P (A) \end{aligned}

The probability of the intersection of an arbitrary number of events is given by

P (⋂_{i = 1}^{n} E_{i}) = P (E_{1}) \cdot P (E_{2} ∣ E_{1}) \cdot P (E_{3} ∣ E_{1} \cap E_{2}) \dots P (E_{n} ∣ E_{1} \cap \dots \cap E_{n - 1})

without loss of generality to the index of $E$ .

P (A ∣ B) = \frac{P (A \cap B)}{P (B)}

Applying this to the right side of the multiplication rule gives

P (E_{1}) \cdot \frac{P (E_{1} \cap E_{2})}{P (E_{1})} \cdot \frac{P (E_{1} \cap E_{2} \cap E_{3})}{P (E_{1} \cap E_{2})} \dots \frac{P (⋂_{i = 1}^{n} E_{i})}{P (⋂_{i = 1}^{n - 1} E_{i})}

such that all terms cancel to give the left side of the multiplication rule.

QED.