 the number of occurrences for the event X, then we have n(aUb)=n(a)+n(b)-n(anb). (1.11) The term that is subtracted in the right-hand side in Eq. (1.11) corresponds to events that would be otherwise counted twice. Two events are said to be mutually exclusive if n(anb) = 0, meaning that the second cannot be observed if the first occurs, and the reverse. In this case, we have the property n(a Ub)=n(a)+ = n(b). As a third definition, the complementary event of x is the event that occurs if x does not occur. It is noted x or sometimes XX. In Fig. 1.4, the complementary event of a Ub, which is noted a Ub, is represented by the shaded surface spreading inside the space S and outside the surface of a Ub. Since any event probability p(x) is proportional to its number of occurrences, X, the previous relation can also be written under the form: p(aUb)=p(a)+p(b)-p(anb), (1.12) which is also usually written as (a+b)=p(a)+p(b)-p(a,b) (1.13) 1.3 Combined, joint, and conditional probabilities where = p(a + b) is the combined probability of the two events a, b and = p(a, b) is the corresponding joint probability (one must be careful to read the + sign in p(a + b) as meaning "or" and not "and"). Finally, the probability p(x) of the complementary event of x (x = a, b, a Ub, anb..) is given by p(x) = 1 - p x . (1.14) Let us apply now the above properties with two illustrative examples. Taking exams Two students, John and Peter, must take an exam. The probability that John passes is p(John pass) = 0.7 and for Peter it is p(Peter pass) = 0.4. The probability that they both pass is p(John pass n Peter pass) = 0.3. What are the probabilities that: (a) At least one of the two students passes the exam? (b) The two students fail the exam? (c) At least one of the two students fails the exam? Answer: In the event (a), we have p(John pass U Peter pass) = p(John pass) + p(Peter pass) - p(John pass n Peter pass) 0.7 + 0.4 - 0.3 = 0.8. The event (b) is the complement of event (a), since the two are mutually exclusive (at least one passing excl