# If $P(ABC)=0.2$, are $A$ and $C$ mutually exclusive?

I was finishing up my statistics homework But I was unsure if I was thinking of the last problem correctly.

If $P(ABC)=0.2$, are $A$ and $C$ mutually exclusive?

My thinking is that mutually exclusive means $A\cap C = \emptyset$ And if that's true $P(ABC)$ would equal $0$, so they are not mutually exclusive.

Thank you.

• What does the symbol "B" mean? – epimorphic Jan 20 '16 at 1:13
• @epimorphic It's just another event, so events $A,B,C$. – Em. Jan 20 '16 at 1:14
• Formatting tips here. – Em. Jan 20 '16 at 1:34

Clearly $ABC\neq \varnothing$. So there exists $x\in ABC$, which means $x$ is a member of $A$ and $B$ and $C$, but if it is a member of $A$ and $C$, then it is a member of $AC$. Therefore $AC\neq \varnothing$, which means $A$ and $C$ are not mutually exclusive.
I think you can also do what you're saying, if you're allowed to know that intersection is commmutative. So, what I mean is, let $P(ABC) = .2$, and suppose $AC=\varnothing$. Then $AC\cap B = \varnothing$. But this gives that $$P(ABC) = P(\varnothing) = 0,$$ which is a contradiction since we know that $P(ABC)=.2$. Thus, it must be the case that $AC\neq \varnothing$. So, $A$ and $C$ are not mutually exclusive.
$ABC \subset AC \implies P(ABC) \leq P(AC)$. Thus, $P(AC) \geq 0.2$ and so $A$ and $C$ are not mutually exclusive.