The theory of probability is a representation of probabilistic concepts in formal terms—that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are interpreted or translated back into the problem domain.
There have been at least two successful attempts to formalize probability, namely the Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation, sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's theorem, probability is taken as a primitive (that is, not further analyzed), and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the rules of probability are the same, except for technical details.
The addition law of probability (aka addition rule or sum rule), states that the probability that A or B will happen is the sum of the probabilities that A will happen and that B will happen, minus the probability that both A and B will happen. The addition rule can be visualized by the equation in Figure 1.
Consider the following example. When drawing one card out of a deck of 52 playing cards, what is the probability of getting a face card (king, queen, or jack) or a heart? Let H denote drawing a heart and F denote drawing a face card. Since there are 13 hearts and a total of 12 face cards (3 of each suit - spades, hearts, diamonds and clubs), but only 3 face cards of hearts, we obtain:
P(H) = 13/52
P(F) = 12/52
P(F ∩ H) = 3/52
Using the addition rule, we get:
The reason for subtracting the last term is that otherwise we would be counting the middle section twice (in case A and B overlap). If A and B are mutually exclusive (also called disjoint, since they do not overlap), then the latter probability is zero.
Addition Rule for Disjoint Events
Therefore, when A and B are mutually exclusive, then P(A ∩ B) = 0, and
P(A ∪ B) = P(A) + P(B), when A ∩ B = ∅.
The symbol ∅ represents the empty set, which means that in this case A and B do not have any elements in common (do not overlap).