alize what has just been described. A set of different events in a family called X may be labeled according to a series X1, X2, XN, where N is the number of events in the event space = {x1, X2, xn}. The probability p(event = Xi), namely, the probability that the outcome turns out to be the event Xi, will be noted p(xi) for short. In the general case, and as we well know, events are neither "absolutely certain" nor "impossible." Therefore, their associated probabilities can be any real number between 0 and 1. Formally, for all events Xi belonging to the space S = {x1, X2, have: 0() 1. (1.2) Probabilities are also commonly defined as percentages. The event is said to have anything between a 0% chance (impossible) and a 100% chance (absolutely certain) of happening, which means strictly the same as using a 0-1 scale. For instance, an election poll will give a 55% chance of a candidate winning. It is equivalent to saying that the odds for this candidate are 55:45, or that p(candidate wins) = 0.55. As a fundamental rule, the sum of all probabilities associated with an event space S is equal to unity. Formally, p(x1)+p(x2) - p(xx)= (1.3) In the above, the symbol (in Greek, capital S or sigma) implies the summation of the argument p(xi) with index i being varied from i = 1 to i = N, as specified under and above the sigma sign. This concise math notation is to be well assimilated, as it will be used extensively throughout this book. We can interpret the above summation rule according to: It is absolutely certain that one event in the space will occur. Probability basics This is another way of stating that the space includes all possibilities, as for the game space defined in Eq. (1.1). I will come back to this notion in Section 1.3, when considering combined probabilities. But how are the probabilities calculated or estimated? The answer depends on whether or not the event space is well or completely defined. Assume first for simplicity the first case: we know for sure all the possible events and the space is complete. C