We have learnt that probabilities are associated with certain events Xi belonging to an event space S = {x1, , X2, XN}. Here, we further develop the analysis by associating events from different subspaces ofS, and establish a new set of rules for the corresponding probabilities. Assume first two subspaces containing a single event, which we note A = {a} and B = {b}, with a, b being included in the space S = {a, b, ...}. Let me now introduce two definitions:5 The combined event is the event corresponding to the occurrence of either a or b or both; it is also called the union of a and b and is equivalently noted a Ub or a V b; The joint event is the event corresponding to the occurrence of both a and b; it is also called the intersection of a and b, and is equivalently noted nb or a coefficient properties can be found in http://mathworld.wolfram.com/BinomialCoefficient.htm or www.math.sdu.edu.cn/mathency/math/b/b219.htm. The symbols U and n are generally used to describe union and intersection of sets, while the symbols V and ^ apply to logical or Boolean variables. If a is an element of the event set, then the Boolean correspondence is "event a = verified or true" For simplicity, I shall use the two notations indifferently. Probability basics Figure 1.4 Venn diagrams showing event space S and its subspaces A and B, with: (a) the intersection An B and (b) the union AUB and its complement AUB. A visual representation of these two definitions is provided by Venn diagrams, as illus- trated in Fig. 1.4. In the figure, the circle S represents the set of all possible occurrences. Its surface is equal to unity. The rectangles A and B represent the set of occurrences for the two events a, b, respectively, and their surfaces correspond to the associated probabilities. The intersection of the two rectangles, An B, defines the number of occurrences (the probability) for the joint event, a nb. The union of the two rectangles, AUB, defines the number of occurrences (the probability) of the combined event, a Ub. If we note n(x),