 x = 7, while there is only one possibility of obtaining either the number x = 2 or the number x = 12. The count of possibilities is shown in the graph in Fig. 1.2(a). Such a graph is referred to as a histogram. If one divides the number of counts by the total number of possibilities (here 36), one obtains the corresponding probabilities. For instance, p(x = 2) = p x = 22) = 1/36 = 0.028, and = 7) = 6/36 = 0.167. The different probabilities are plotted in Fig. 1.2(b). To complete the plot, we have 1.1 Events, event space, and probabilities 23456 345678 Figure 1.1 The 36 possible outcomes of counting points from casting two dice. 0.180 0.160 0.140 0.120 0.100 0.080 0.060 0.040 0.020 0.000 Count Figure 1.2 (a) Number of possibilities associated with each possible outcome of casting two dice, (b) corresponding probability distribution. included the two count events X = 1 and x = 13, which both have zero probability. Such a plot is referred to as the probability distribution; it is also called the probability distribution function (PDF). See more in Chapter 2 on PDFs and examples. Consistently with the rule in Eq. (1.3), the sum of all probabilities is equal to unity. It is equivalent to say that the surface between the PDF curve linking the different points (x, (p(x)) and Probability basics the horizontal axis is unity. Indeed, this surface is given by S = (13 = 7)/2 12*(6/36)/2=1. The last example allows us to introduce a fundamental definition of the probability (xi) in the general case where the events Xi in the space S = {x1, x2 XN} do not have equal likelihood: of possibilities for event (1.4) of possibilities for all events This general definition has been used in the three previous examples. The single coin tossing or single die casting are characterized by equiprobable events, in which case the PDF is said to be uniform. In the case of the two-dice roll, the PDF is nonuniform with a triangular shape, and peaks about the event X = 7, as we have just seen. Here we are reaching a subtle point in the notion of p