Q:

Find the probability of an event given the probability of the union and the probability of the complement

"There are two independent events, A and B, and we know that the probability that both occurs simultaneously is $\frac{1}{6}$ and the probability that none of them occurs is $\frac{1}{3}$. So, what is the probability that just one of them occurs?"
I need to solve this assignment's question and I'm struggling to figure out how to do that. I think I need to know P(A$\bigcap$B) to solve it but I cannot find that!
Does anyone know how to solve it?

A:

Case I. You only need to know the probability that one of the two events occurs, not the two actual probabilities.
In that case, if both occur with probability $\frac{1}{6}$, and neither occurs with probability $\frac{1}{3}$, then the only remaining possibility is that exactly one occurs, and that must be what?
Case II. You actually need to determine the two probabilities $P(A)$ and $P(B)$ (though you can't know which is which).
The fact that the two events are independent means that the probability that both occur simultaneously can be written as $P(A)P(B)$.  You know that equals $\frac{1}{6}$.  The probability that neither occurs is $[1-P(A)][1-P(B)]$, and you know that equals $\frac{1}{3}$.
You should now be able to solve for the two probabilities, although you can't tell which is which.

