atively by computing n = log(0.5)/log(364/365)). As expected, this is a relatively large group. Question (c) leads to a nonintuitive result, as we shall see. The number of ways of selecting a pair of persons in a group of n people is C2 = n(n - 1)/2. The probability that two people do not share birthdays is 364/365. Thus, the probability of finding no matching pair in the whole group is p(x) = (364/365)"(n-1)/2, and the probability of finding at least one matching pair is p(x) = 1 - p(x) - (364/365)"("-1)/2. Solving for p(x) 22 0.5 (or n(n - 1)/2 = 253) yields n = 23. Thus, a group as small as 23 people has a 50% chance of including at least two persons sharing the same birthday. Such a result is indeed far from intuitive! I address next the concept of conditional probabilities. By definition, the conditional probability p(b |a) is the probability that event b occurs given the fact that event a has occurred. The factual knowledge regarding the first event thus provides a clue regarding the likelihood of the second. The conditional probability is calculated according to what is called Bayes's theorem: p(b)a)= (1.15) 6 For simplicity, we excluded here people born on February 29th (which only happens every four years). It is also assumed that birthday events are independent, which is not true in real life, for instance, due to seasonal peaks (i.e., observed peaks in the summer months) and effects of induced labor (i.e., a majority of births happen on weekdays). A full analysis of the different ways to analyze and solve the Birthday paradox can be found in http://en.wikipedia.org/wiki/Birthday_paradox 1.3 Combined, joint, and conditional probabilities Alternatively, we can write p(a,b)=p(ba)p(a)=p(ab)p(b), = (1.16) which defines the fundamental relation between joint and conditional probabilities. there is no correlation whatsoever between two events a and b, the events are said to be uncorrelated or independent. This means that the probability of b knowing a (or any other event different from b) is unchanged, which 