Probability Chain Rule The application of the Product Rule to a set of random variables via their joint probabilities P(X1,…,Xn)= P(Xn∣Xn−1,…,X1)⋅P(Xn−1∣Xn−2,…,X1)⋅…⋅P(X2∣X1)⋅P(X1)\begin{align*} \mathrm{P}\left(X_1, \ldots, X_n\right) = & \text{ }\mathrm{P}\left(X_n \mid X_{n-1}, \ldots, X_1\right) \cdot \mathrm{P}\left(X_{n-1} \mid X_{n-2}, \ldots, X_1\right) \cdot \ldots \\ & \cdot \mathrm{P}\left(X_2 \mid X_1\right) \cdot \mathrm{P}\left(X_1\right) \end{align*}P(X1,…,Xn)= P(Xn∣Xn−1,…,X1)⋅P(Xn−1∣Xn−2,…,X1)⋅…⋅P(X2∣X1)⋅P(X1)