I have a question, so in my textbook they give us this definition:
So say we have three events: $A$, $B$, and $C$.
If I wanted to find $P(A \cup B \cup C)$, would I do $P(A) + P(B) + P(C)$?
Because don't we also have to subtract the intersection too?
$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A\cap B) - P(A\cap C) - P(B\cap C) + P(A\cap B\cap C)$
This was a "generalization" our Professor went over. But how do I know when to either only add up the probability of the events vs. adding them up and then subtracting the intersections of two, and then adding the intersections of all 3?
If the events are mutually exclusive then the probability of A AND B, A AND C, B AND C, A AND B AND C is zero. Remember that intersection represents a logical AND. If the events don't influence each other then the probability of them happeneing together is zero.
In notation this would look like:
$P(A \cup B \cup C) = P(A) + P(B) + P(C) - 0 - 0 - 0 + 0 = P(A) + P(B) + P(C)$