possible birthdays, and that each person's birthday is equally likely to be any of these days, independent of the other people in the group. For independent birthdays, the uniform distribution on birthdays is the distribution that minimizes the probability of two people with the same birthday; any unevenness increases this probability. The problem of a non-uniform number of births occurring during each day of the year was first addressed by Murray Klamkin in 1967. As it happens, the real-world distribution yields a critical size of 23 to reach 50%. The goal is to compute , the probability that at least two people in the room have the same birthday. However, it is simpler to calculate , the probability that no two people in the room have the same birthday. Then, because and are the only two possibilities and are also mutually exclusive, Here is the calculation of for 23 people. Let the 23 people be numbered 1 to 23. The event that all 23 people have different birthdays is the same as the event that person 2 does not have the same birthday as person 1, and that person 3 does not have the same birthday as either person 1 or person 2, and so on, and finally that person 23 does not have the same birthday as any of persons 1 through 22. Let these events be called Event 2, Event 3, and so on. Event 1 is the event of person 1 having a birthday, which occurs with probability 1. This conjunction of events may be computed using conditional probability: the probability of Event 2 is , as person 2 may have any birthday other than the birthday of person 1. Similarly, the probability of Event 3 given that Event 2 occurred is , as person 3 may have any of the birthdays not already taken by persons 1 and 2. This continues until finally the probability of Event 23 given that all preceding events occurred is . Finally, the principle of conditional probability implies that is equal to the product of these individual probabilities: The terms of equation () can be collected to arrive at: Evaluating equation () gives Therefore, (50.7297%). This process can be generalized to a group of people, where is the probability of at least two of the people sharing a birthday. It is easier to first calculate the probability that all birthdays are