# Events A and B are independent, events A and C are mutually exclusive, and events B and C are independent…

I couldn't fit all of the problem in the title, but here it is in full:

Events A and B are independent, events A and C are mutually exclusive, and events B and C are independent.

If P(A) = $\frac{1}{2}$; P(B) = $\frac{1}{4}$; P(C) = $\frac{1}{8}$; what is P($A \cup B \cup C$)?

I'm a little lost on how to approach this - I understand the difference between mutually exclusive and independent events (being that mutually exclusive events both cannot happen at once, and independent events do not affect one another) but I don't entirely understand what formula I should be using to figure out how to solve the problem.

Note that the event $A\cup B\cup C$ can happen in the following two disjoint ways: (i) $B$ holds or (ii) $B$ fails and one of $A$ or $C$ holds.
The probability of (i) is $1/4$.
For probability of (ii) we need to work somewhat harder. Since $A$ and $C$ are mutually exclusive, we want $\Pr(B'\cap A)+\Pr(B'\cap C)$. Here $B'$ denotes the complement of $B$. The required probabilities can be found by independence.
• You are welcome. A Venn diagram that breaks things up into all the basic ways the event can happen, like $A\cap B\cap C'$ and so on can be used, I tried to do the calculation reasonably efficiently. But the analysis came from a Venn Diagram visualization, done in the head because I can't draw and type at the same time. – André Nicolas Sep 22 '15 at 0:08
• Here is the visualization. Two ovals, nicely separated, representing $A$ and $C$. A nice fat circle in the middle, representing $B$ and overlapping with $A$ and with $C$. I want $B$ plus the two outer ears. – André Nicolas Sep 22 '15 at 0:16