udes both failures). Therefore p(John fail n Peter fail) = 1 - p(John pass U Peter pass) = 1 - 0.8 = 0.2. The probability of the event (c) can be calculated as follows: p(John fail U Peter fail) = p(John fail) + p(Peter fail) - p(John fail n Peter fail) = (1 - - 0.4) -0.2=0.7. To conclude this example, we should take note of an interesting feature. Indeed, we p(John pass) U Peter pass = >p(John pass) X p(Peter pass) = 0.7 X 0.3 = 0.21, which means that the probability of both students passing the exam is greater than the product of their respective probabilities of succeeding. This means that the events of their individual successes are correlated. This notion of event correlation will be clarified further on. Probability basics Sharing birthdays Given a group of people, the issue is to find how many persons may share your birthday (assuming that birth events are independent): (a) In a group of n people, what is the probability of finding at least one person who shares your birthday? (b) How many people n should be in the group SO that this probability is 50%? (c) How many people n should be in the group SO that the probability to find at least two persons sharing the same birthday is 50%? Answer: The probability that any person met at random shares your birthday is 1/365 (since your birthday is one date out of 365 possibilities6, which is relatively small (1/365 = 0.27%). The probability that this person does not share your birthday is thus 364/365 = 99.7%, which is relatively high, as expected. To answer question (a), we consider the complementary event x = nobody in this group shares your birthday. The probability that two persons selected at random do not share your birthday is (364/365)2, thus for a group of n people we have p(x) = (364/365)". The probability that at least one person shares your birthday is, therefore, p(x) = 1 - p(x) = 1 - (364/365)". Question (b) is answered by finding the value of n for which p(x) 2 0.5. With a pocket calculator, one easily finds after a few trials that n = 253 (or altern