Chain rule for probabilities ($P(A_1\cap \cdots \cap A_m)={\prod }_{j}P(A_j\mid A_1\cap \cdots \cap A_{j-1})$)

When all conditioning events have positive probability, repeatedly apply the definition of conditional probability to factor a joint event into successive conditional factors. For a sequence of discrete responses, this yields the product of conditional masses along the actual outcome path. It requires no unconditional independence assumption.