Calculating probabilities of elementary events

Probability distribution

Let \ohm be the sample space, then P: \ohm \to [0, 1] a probability distribution, if the following holds: \sum_{w\in\ohm} P(w) = 1

If P(w)=0, then is w a impossible outcome. If P(w)=1, then is w a safe outcome. Each w is refereed to as an elementary event

For a outcome A \subseteq \ohm is \bar{A}=\ohm \setminus A the complement and the following holds: P(\bar{A}) = 1 - P(A)

Calculations rules for probabilities

Let \ohm be the sample space and Pr: \ohm \to [0, 1] a probability distribution, then the following calculation rules apply:

1. P(\ohm = 1), P(\emptyset)=0 and \forall A \subseteq \ohm : P(A) \geq 0
2. P(A \cup B) = P(A) + P(B) - P(A \cap B) \le P(A) + P(B)
3. A \cap B = \emptyset \implies P(A \cup B) = P(A) + P(B)
4. P(A\cap B) \geq P(A) - P(\bar{B})
5. P(A \setminus B) = P(A) - P(A \cap B)