Question: I'm taking a graduate course in probability and statistics using [Larsen and Marx, 4th edition](http://rads.stackoverflow.com/amzn/click/0131867938) and I'm struggling with a seemingly basic question.

If A and B are any two events, not mutually exclusive: $$P((A \cup B) ^\complement) = 0.6, P(A \cap B) = 0.2$$

What is the probability that A or B occurs, but not both? Or in other words: $$P((A \cap B)\ ^\complement \cap (A \cup B)) = ?$$

So far, I've been able to infer the following: $$ P(A \cup B) = 1 - P((A \cup B) ^\complement ) = 1 - 0.6 = 0.4 $$ and $$P((A \cap B) ^\complement) = 1 - P(A \cap B) = 1 - 0.2 = 0.8$$

Can someone kindly give a hint as to how to approach from here? Am I heading in the right direction? How should I think about these types of problems in general? I feel like the text gives you the basic set of axioms to define things but then neglects the showing of solutions for slightly more complicated examples of compound probability equations such as the above.

Answer: 
**Hint:** Find the probability of $A$ only plus the probability of $B$ only.

Let $x= P(A~ \text{only})$, $y=P(B ~\text{only})$. Then $x +y + 0.2 =0.4$. Thus $x+y= 0.2$
THe probability that at least one of them occurs is $0.4$ and the probability that both occur is $0.2$. Thus, the probability that *exactly* one occurs is $0.4-0.2=0.2$.