How do I prove this probability question?

Question

Show that if $A\subset B\subset C\subset D$, then

$P(A\cap B\cap C\cap D)=P(A/B) \cdot P(B/C)\cdot P(C/D)\cdot P(D)$

Answer 1

• $P(C\mid D)P(D)=P(C\cap D)=P(C)$ since $C\subseteq D$ implies that $C=C\cap D$.
• $P(B\mid C)P(C)=P(B\cap C)=P(B)$ since $B\subseteq C$ implies that $B=B\cap C$.
• $P(A\mid B)P(B)=P(A\cap B)=P(A)$ since $A\subseteq B$ implies that $C=C\cap D$.
• $P(A)=P(A\cap B\cap C\cap D)$ since $A\subseteq B\subseteq C\subseteq D$ implies that $A=A\cap B\cap C\cap D$.