My stats prof posted an answer but it doesn't make sense. Please help!
$A$, $B$, $C$ are mutually independent. Prove $A$ and $B^c ∪ C$ are independent? ($B^c$ = $B$ complement)
My stats professor posted this answer, but I don't get where he got the first line in the proof. Shouldn't it be $P(A \cap (B^c \cup C))$? Would really appreciate it if anyone could explain. Thank you so much!
Proof: $$P(A∪(B^c ∩C))=P((A∩B^c)∪(A∩C)) $$
$$= P(AB^c) + P(AC) − P(AB^cC) $$
$$= P(A)P(B^c) + P(A)P(C) − P(A)P(B^c)P(C),\text{ because they are mutually independent.}$$
$$= P(A)(P(B^c) + P(C) − P(B^cC)) $$
$$=P(A)P(B^c ∪C) $$
Thus, $A$ and $B^c ∪ C$ are independent.
I agree, the first line should say $P(A \cap (B^C \cup C))$. However, let's assume we have that instead. Then, we can distribute the $A\cap$ over $B^C \cup C$ to get $P((A \cap B^C) \cup (A \cap C))$. This is exactly where the professor went, so from then on, the proof is correct. It is just a typo in the first line; everything else in the proof is correct.