When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event. The same thing we could say about sum of A, B and C.
In several articles I found another definition of the sum:
When two events, A and B, are mutually exclusive, the probability that A or B or A and B will occur is the sum of the probability of each event.
First question: Is second definition correct?
For second question let imagine the following example:
Probability of getting tail of the coin is 1/2. Person throws coin 3 times.
- Event A: get tail at first throw, P(A) = 1/2
- Event B: get tail at second throw, P(B) = 1/2
- Event C: get tail at third throw, P(C) = 1/2
Then probability of getting tail at first or second or third throw is:
P(A + B + C) = 1/2 + 1/2 + 1/2 = 3/2, but 3/2 > 1
But as we know probability P of event is always 0 <= P <= 1
Second question: Where am I wrong?
First question: Yes, the definition is correct. If $A$ and $B$ are mutually exclusive, then: $$P(A \cap B)=0$$ and: $$P(A \cup B) = P(A) + P(B)$$ and so indeed we have that: $$P(A \cup B \cup (A \cap B))=P(A) + P(B)$$
Second question: Your events are not mutually exclusive, and so you cannot add up the probabilities: you can only use $$P(A \cup B) = P(A) + P(B)$$ when $A$ and $B$ are mutually exclusive. And your events are not mutually exclusive because any two of them can happen in the same scenario. In fact, they can all three happen: you throw a tail for all three tosses.