Prove that $A, B, C$ are independent (under certain conditions) Suppose:


*

*$A$ is independent from both $B\cap C$ and $B \cup C$.

*$B$ is independent from $A \cap C$.

*$C$ is independent from $A \cap B$.

*$P(A), P(B), P(C)$ are all greater than zero.
Prove that $A, B, C$ are independent.
I know I have to show that the 3 events are all pairwise independent and that $P(A\cap B \cap C) = P(A) P(B) P(C)$.
Working with 1., I got to the equality $P(A \cap B) + P(A \cap C) = P(A) [P(B\cap C) + P(B \cup C)]$, but I don't find it very useful. I tried several other approaches and they didn't work.
Could someone give me any hints? I'd also appreciate any intuition into how to solve these type of exercises. I always solve them by trial and error. Thanks!
 A: I think you are on the right track.
Let us start from the equation you got from hypothesis 1.
\begin{eqnarray}
P(A\cap B) &=& P(A) \cdot P(B\cup C) + P(A)\cdot P(B\cap C)-P(A\cap C)=\\
&=&P(A) [P(B) + P(C) - P(B\cap C)]+ P(A)\cdot P(B\cap C)-P(A\cap C)=\\
&=& P(A)\cdot P(B) + P(A)\cdot P(C) -P(A\cap C).
\end{eqnarray}
So we have
\begin{equation}
P(A\cap B) -P(A)\cdot P(B) = P(A)\cdot P(C) - P(A\cap C).\tag{1}\label{1}
\end{equation}
Now condition 4. makes me think that I could multiply both sides of \eqref{1} by $P(B)\cdot P(C)$
and get
\begin{equation}
P(B)\cdot P(C)\cdot P(A\cap B) -P(A)\cdot P^2(B)\cdot P(C) =\\ =P(A)\cdot P(B)\cdot P^2(C) - P(B)\cdot P(C)\cdot P(A\cap C).\tag{2}\label{2}
\end{equation}
By hypothesis 2. and 3.
\begin{equation}
P(A\cap B\cap C) = P(B) \cdot P(A\cap C) = P(C) \cdot P(A\cap B),\tag{3}\label{3}
\end{equation}
so that \eqref{2} yields
\begin{equation}
P(B) \cdot P(A\cap B\cap C) -P(A)\cdot P^2(B)\cdot P(C) =\\
=P(A)\cdot P(B)\cdot P^2(C)-P(C) \cdot P(A\cap B\cap C),
\end{equation}
that is
\begin{equation}
[P(B) +P(C)]\cdot [P(A\cap B\cap C) -P(A)\cdot P(B)\cdot P(C)] = 0.
\end{equation}
From the last equation you get
\begin{equation}
P(A\cap B\cap C) =P(A)\cdot P(B)\cdot P(C)\tag{4}\label{4}.
\end{equation}
Use now \eqref{4} and \eqref{3} to get pairwise independence.

As for the general intuition, my suggestion is to always ask yourself: "did I use all the hypotheses?", "How can I translate into mathematical terms the hypotheses I have not yet used?" Note also here, the hidden use of the expression relating probability of union and probability of intersection.
