Looking for a bit of help here. If $B$ and $C$ are disjoint, and $D = B^c \cap C^c$ does that make D=1 and therefore they would form a partition? I think it cannot be that simple.
The full question also includes the following. Events $A,B,C,D$ where $P (A\mid B)= P(A\mid C)= P(A\mid D)= 0.5$. $B \cap\ C$ = $\phi\ $ and $D = B^c \cap C^c$
Find P(A).
Not really sure where to start.
Let $X$ be the whole set, then $B,C$ and $D$ form a disjoint partition of $X$. From this it follows that $$P(A)=P(B)P(A|B)+P(C)P(A|C)+P(D)P(A|D)=\frac{1}{2}(P(B)+P(C)+P(D))=\frac{1}{2}.$$