I have a colleague who wishes to compute, for 4 events, $$ P(A \cap B \cap C \cap D ) \tag1 $$ I believe this is $$ P(A)P(B|A)P(C|A\cap B)P(D|A \cap B \cap C) \tag2 $$ They know $P(A), P(B), P(C), P(D)$ and it turns out that $P(D|A \cap B \cap C)=1$. They do not know $P(C|A\cap B)$ or $P(B|A)$ so they are instead using $$ P(A)P(B)P(C)1 \tag3 $$ as a surrogate for $(2)$.
I believe this is incorrect.
[1] Is $(2)$ correct as an expression for $(1)$?
[2] I believe it is a bad idea but is using $(3)$ a bad idea as an approximation for $(2)$?
[3] My colleague is quite convinced $(3)$ is ok in place of $(2)$, so much so I am writing for reassurance! How can I convince my colleague it's not ok, or am I wrong?
Thanks in advance.
EDIT & CLARIFICATION
Apologies @drhab, for forgetting to mention this, the events are not independent.
[1] Yes, (2) is correct.
[2] (3) is in general sense incorrect. In special case where $A,B,C$ are independent it is correct.
[3] Attend your colleague on the possibility $A=B=C$ (or if you dislike equalities in this matter a case with high level of dependence). In that case (2) gives $P(A)$ as solution (which is correct) and (3) gives $P(A)^3$ which is incorrect if $P(A)\notin\{0,1\}$.