# Math Help - Proof union disjoint events

1. ## Proof union disjoint events

Proofs scare the living daylights out of me! I could really use some help with this:

For some experiment, suppose B and C are events. The event, D, that "exactly one of the events B and C occur" is of interest.
(a) Show that D can be written as a union of disjoint events.
(b) Show that P(D) = P(B) + P(C) -2P(B intersect C).

Thanks!

2. (a)

$D = (B \setminus C) \cup (C \setminus B)$

(b)

We can use the above decomposition to do that, however I think it is easier to do this other way.
We will use the formula, valid for any events A,B:
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Hence

$P(D) = P((B \cup C) \setminus (B \cap C)) = P(B \cup C) - P(B \cap C) = P(B) + P(C) - 2P(B \cap C)$