"Whoops! One last point on the topic of independence: please don’t make the mistake of thinking that mutually exclusive events are independent! This is by no means the case, and in fact, the opposite is true. If two events are mutually exclusive, they are extremely dependent on each other! Consider the most trivial case: I choose a random person on campus, and define I as the event that they’re an in-state student, and O as the event that they’re out-of-state. Clearly these events are mutually exclusive. But are they independent? Of course not! Think about it: if I told you a person was in-state, would that tell you anything about whether they were out-of-state? Duh. In a mutually exclusive case like this, event I completely rules out O (and vice versa), which means that although Pr( I ) might be .635, Pr( I | O ) is a big fat zero. More generally, Pr( A | B ) is most certainly not going to be equal to Pr( A ) if the two events are mutually exclusive, because learning about one event tells you everything about the other."