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.


1 Answer 1


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.

It remains to put the pieces together.

  • $\begingroup$ Thanks! This made things a bit easier, I was attempting to draw venn diagrams of the problem and it wasn't helping $\endgroup$
    – secondubly
    Sep 22, 2015 at 0:00
  • $\begingroup$ 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. $\endgroup$ Sep 22, 2015 at 0:08
  • 1
    $\begingroup$ 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. $\endgroup$ Sep 22, 2015 at 0:16

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