Probability question on mutual exclusiveness

Question

Okay, it's like this:

Let A, B and C be events such that P[A|C] = 0.05 and P[B|C] = 0.05.

I was just wondering, should the events A and B be looked at as mutually exclusive, as in only one or the other could happen? In the back of my book they calculated P[A U B|C] as:

P[A|C] + P[B|C]. I don't really understand the justification. I mean, who's to say A and B can't happen at the same time?

EDIT:

Let A, B and C be events such that P[A|C] = 0.05 and P[B|C] = 0.05. Which of the following statements is true.

A) P[A n B|c] = (0.05)^2

B) P[A' n B'|C] >= .90

C) P[A U B|C] <= 0.05.

D) P[A U B|C'] >= 1 - (0.05)^2

E) P[A U B|C'] >= .10

Answer 1

You're quite right, unless the book supplied additional information. In fact A and B could be the very same event.

Answer 2

Definition of two events being mutually exclusive: $$ P(A \cup B) = P(A) +P(B) $$ i.e., the event $P(A \cap B)=0$. If in your case $P(A \cup B|C) = 0.1$ then yes, $P(A|C)$ and $P(B|C)$ are mutually exclusive.