nd is a little over 50%, which was not an intuitive result. The lesson learnt is that successive events may be independent, but their respective probabilities keep evolving according to the outcome of the preceding events. Put otherwise, each occurring event affects the space of the next series of events, even if all events are 100% independent. To complete this first chapter, I must introduce the summing rule of conditional probabilities, also called the law of total probability. Assume two complete and disjoint event spaces S = {a, b} and T = {x, y, z}, with S O T = (the symbol meaning an empty set or a set having zero elements) and with, by definition of probabilities p(a) + p(b) = 1, and p(x) + p(y) + p(z) 1. Then the following summing relations hold: p(x)=p(xa)p(a)+p(xb)p(b) (1.18) p(y)=p(ya)p(a)+p(yb)p(b) p(ax)p(x)+p(aly)p(y)+p(az)p(z) (1.19) p(b)=p(bx)p(x)+p(by)p(y)+p(bz)p(z). Probability basics Thus with the knowledge of the conditional probabilities p(.).) relating the two event space causalities, one is able to compute the probabilities from one space to the other.7 It is important to note that while the conditional p(.1.) are true probabilities, they do not sum up to unity, a mistake to avoid by firmly memorizing any of the above summing relations. Exercises 1.1 (B): Flipping two coins simultaneously, what are the probabilities associated with the following events: (a) Getting two heads? (b) Getting one heads and one tails? (c) Getting either two heads or two tails? 1.2 (B): Rolling three dice, one wins if the outcome is 4, 2, 1 in any order. What is the probability of winning in the first, the second, and the third dice roll? What is the number of rolls required to have at least a 50% chance of winning? 1.3 (B): A lotto game has 50 numbered balls, out of which six are picked at random. What is the probability of winning by betting on any six-number combination? 1.4 (B): Three competing car companies, A, B, and C, have market shares of 60%, 30%, and 10%, respectively. The probabilities that the cars wi