Why doesn't $P(A\cup B \cap C) = P(A) + P(B\cap C) - P(A\cap B \cap C)$? 
Why doesn't $P(A\cup B \cap C) = P(A) + P(B\cap C) - P(A\cap B \cap C)$?

I understand that this isn't true looking at a Venn diagram, but mathematically it doesn't make sense if $P(A\cup B) = P(A) + P(B) - P(A\cap B)$.
 A: It is true that $$P(X\cup Y) = P(X)+P(Y)-P(X\cap Y).\tag{1}$$
And if we substitute $X=A, Y=B\cap C$ blindly, that would seem to mean:
$$P(A\cup B\cap C) = P(A)+P(B\cap C) - P(A\cap B\cap C).$$
But that is an improper substitution. In the text, we have to substitute $(B\cap C)$ into the formula for $Y.$ 

A numeric example
We know that $3\times 2 = 6$ and that $2=1+1.$ But we can't just replace the string $2$ in the first expression with the string $1+1$ and get:
$$3\times 1+1 = 6\quad\color{red}{\text{wrong}}$$
We have to replace $2$ by $(1+1)$ to get:
$$3\times(1+1) = 6\quad \color{green}{\text{correct}}$$

So it is true that, from $(1)$, we get:
$$P(A\cup(B\cap C))=P(A)+P(B\cap C) - P(A\cap B\cap C)$$
but it is not true that:
$$P((A\cup B)\cap C) =P(A)+P(B\cap C) - P(A\cap B\cap C)$$
and $A\cup B\cap C$ is normally interpreted as $(A\cup B)\cap C,$ and it is not, in general, true that:
$$(A\cup B)\cap C=A\cup (B\cap C).$$

The standard notation we use, where we put operators like $+$ and $\times$ and $\cap$ and $\cup$ in between operands is called "infix notation." Infix notation does not play well with string substitution. 
Another notation, reverse Polish notation, actually does okay with string substitutions. If we write $3,2,\times = 6$ and $2=1,1,+$ then we can replace the $2$ in the first expression with $1,1,+$ and still get a valid equality:
$$3,1,1,+,\times=6$$
(I  use infix $=$ because $=$ is not a operator, but we could be more pedantic.)
